To Settle Infinity Question, a New Law of Mathematics (2013)(quantamagazine.org)
quantamagazine.org
To Settle Infinity Question, a New Law of Mathematics (2013)
https://www.quantamagazine.org/to-settle-infinity-question-a-new-law-of-mathematics-20131126/
66 comments
Great post, let me just add one point:
> The 21-century understanding is something like: there's no need for there to be "ultimate" foundations.
The modern view is that there simply is no ultimate foundation. Large cardinal axioms in set theory can be seen as adding more and more inner models of set theory (with fewer large cardinals) into an ambient theory. This is basically the same thing as asserting that "the previous theory is consistent". You can play this game forever and it will not converge - the result hinted at in the article states something different.
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In the end we use set theory as an alibi. We tell people that ultimately all of mathematics can be encoded in set theory, but it is neither a natural encoding, nor is it really true since we are talking about different flavors of "set theory".
Algebraic geometry is a good example. To begin with, you are not using ZFC to encode categories - since you need to be able to manipulate proper classes as if they were sets - so we need to use some extension such as NBG set theory instead. Then, in order to construct the localisation of a large category you suddenly need large cardinals, even though intuitively the localisation is in some sense no larger than the category you start with.
And this is the point where we forget this construction again, because it doesn't give any useful insights.
> The 21-century understanding is something like: there's no need for there to be "ultimate" foundations.
The modern view is that there simply is no ultimate foundation. Large cardinal axioms in set theory can be seen as adding more and more inner models of set theory (with fewer large cardinals) into an ambient theory. This is basically the same thing as asserting that "the previous theory is consistent". You can play this game forever and it will not converge - the result hinted at in the article states something different.
---
In the end we use set theory as an alibi. We tell people that ultimately all of mathematics can be encoded in set theory, but it is neither a natural encoding, nor is it really true since we are talking about different flavors of "set theory".
Algebraic geometry is a good example. To begin with, you are not using ZFC to encode categories - since you need to be able to manipulate proper classes as if they were sets - so we need to use some extension such as NBG set theory instead. Then, in order to construct the localisation of a large category you suddenly need large cardinals, even though intuitively the localisation is in some sense no larger than the category you start with.
And this is the point where we forget this construction again, because it doesn't give any useful insights.
"We do not nowadays refute our predecessors, we pleasantly bid them good-bye."
Here's Woodin talking about his work in this area on a more technical level: https://www.youtube.com/watch?v=nVF4N1Ix5WI
I don't really understand this. "Infinity" just means without finite quantity, doesn't it? I can, theoretically, count for infinity - there are no end to the theoretical numbers.
If I can prove a value can be reduced arbitrarily close to a limit, we can say something is equal to that limit for any finite resolution (for any resolution of "closeness to the limit" there is a precision in the calculation capable of delivering within that resolution)
Hence, I can say "F(x) = 3 for any finite resolution", so if no infinities exist, then none are needed for mathematics. They are a calculation artifact that represents a single function over arbitrary levels of resolution.
If I can prove a value can be reduced arbitrarily close to a limit, we can say something is equal to that limit for any finite resolution (for any resolution of "closeness to the limit" there is a precision in the calculation capable of delivering within that resolution)
Hence, I can say "F(x) = 3 for any finite resolution", so if no infinities exist, then none are needed for mathematics. They are a calculation artifact that represents a single function over arbitrary levels of resolution.
This is crazy. How about just doing what maths does anyways - state your assumptions?
Prove stuff with V=ultimate L? Great, slap a "V = ultimate L => ..." on it.
Prove stuff with forcing axiom? Cool, "forcing axiom implies ..."
The entire premise of asking "whats the one true axiom?" makes no sense. Its not like the real world places any restrictions on math - its a self-contained universe. So obviously you are free to pick whatever imaginary universe you would like to study. If we ever get to a point where we can directly test which of the 2 axioms leads to more useful theories in the real world, great, but that seems pretty unlikely given they come up when talking about infinity. The obvious truth seems to be that infinity could have a few subtly different meanings depending on how you define it - and thats totally fine because the real world does not contain infinities, its a purely mental construct.
Prove stuff with V=ultimate L? Great, slap a "V = ultimate L => ..." on it.
Prove stuff with forcing axiom? Cool, "forcing axiom implies ..."
The entire premise of asking "whats the one true axiom?" makes no sense. Its not like the real world places any restrictions on math - its a self-contained universe. So obviously you are free to pick whatever imaginary universe you would like to study. If we ever get to a point where we can directly test which of the 2 axioms leads to more useful theories in the real world, great, but that seems pretty unlikely given they come up when talking about infinity. The obvious truth seems to be that infinity could have a few subtly different meanings depending on how you define it - and thats totally fine because the real world does not contain infinities, its a purely mental construct.
The article frames it as some sort of debate in some sort of mathematical ruling body which doesn't exist. There is no authority that's going to decide that, "ok, V = ultimate L is the axiom we're going with, everyone has to accept it now."
If you're assuming ZFC, you typically don't say anything (although I personally like to explicitly point out if I need the axiom of choice.) There was no ruling body that decided, "ok, we're all using ZFC now." Earlier generations of mathematicians widely adopted it, and now it's just the default.
Fun fact! Some recently popular mathematics leaking into the realm of CS, category theory, is usually not phrased in the language of ZFC. Instead, category theory is easier to formulate in NBG set theory, which is equivalent to ZFC in the sense that a ZFC statement is provable in NBG if and only if it is provable in ZFC.
The advantages afforded by NBG are:
1. the ontology is bigger. NBG has classes and sets, ZFC just has sets, and
2. it is finitely axiomatizable!
Unlike what the linked article says, ZFC does NOT have finitely many axioms! ZFC has 7 axioms and 2 axiom schemas, which are sort of meta-axioms that each generate countably infinite axioms.
On the other hand, ZFC is a bit more explicit and hands on. It's easier to roll up your sleeves and just get elbows deep in sets, which is why it's favored more by actual set theorists and set-theoretic topologists.
If you're assuming ZFC, you typically don't say anything (although I personally like to explicitly point out if I need the axiom of choice.) There was no ruling body that decided, "ok, we're all using ZFC now." Earlier generations of mathematicians widely adopted it, and now it's just the default.
Fun fact! Some recently popular mathematics leaking into the realm of CS, category theory, is usually not phrased in the language of ZFC. Instead, category theory is easier to formulate in NBG set theory, which is equivalent to ZFC in the sense that a ZFC statement is provable in NBG if and only if it is provable in ZFC.
The advantages afforded by NBG are:
1. the ontology is bigger. NBG has classes and sets, ZFC just has sets, and
2. it is finitely axiomatizable!
Unlike what the linked article says, ZFC does NOT have finitely many axioms! ZFC has 7 axioms and 2 axiom schemas, which are sort of meta-axioms that each generate countably infinite axioms.
On the other hand, ZFC is a bit more explicit and hands on. It's easier to roll up your sleeves and just get elbows deep in sets, which is why it's favored more by actual set theorists and set-theoretic topologists.
Can you unpack 2 for me? How can there only be finitely many axioms if NBG is 'equivalent' in proving power to ZFC?
Unfortunately I'm not a set theorist, so I haven't done any deep dives into the NBG/ZFC equivalence. My area of expertise (Lie theory) doesn't really require detailed knowledge along those lines. I'm only really aware of it because I'm a category theory enthusiast.
That said, my intuition is that NBG is finitely axiomatizeable with equivalent strength precisely because of #1: it has a bigger ontology.
From what I remember of grad school set theory, working with proper classes in ZFC amounts to clunky manipulations of formulas in the language of set theory, because there is no such thing as a class in ZFC. A proper class is a well-formed formula that fails to describe a set. It can be described in the language, but cannot be constructed by the axioms. Axiom schemas give one axiom for all formulas in the language of ZFC.
Meanwhile, classes are an actual object in the language of NBG, so an axiom of NBG can just say "for all classes x..."
There's a wikipedia article[1] that actually goes into a good bit of detail in the case of the axiom schema of specification.
[1] https://en.wikipedia.org/wiki/Axiom_schema_of_specification
That said, my intuition is that NBG is finitely axiomatizeable with equivalent strength precisely because of #1: it has a bigger ontology.
From what I remember of grad school set theory, working with proper classes in ZFC amounts to clunky manipulations of formulas in the language of set theory, because there is no such thing as a class in ZFC. A proper class is a well-formed formula that fails to describe a set. It can be described in the language, but cannot be constructed by the axioms. Axiom schemas give one axiom for all formulas in the language of ZFC.
Meanwhile, classes are an actual object in the language of NBG, so an axiom of NBG can just say "for all classes x..."
There's a wikipedia article[1] that actually goes into a good bit of detail in the case of the axiom schema of specification.
[1] https://en.wikipedia.org/wiki/Axiom_schema_of_specification
In most fields of mathematics, it is not necessary to state that ZFC is assumed. Axiom of choice is singled out sometimes, but the other axioms are considered so self-evident that it's not even worth mentioning. Yet from a logical standpoint, there's nothing special about ZFC to be so universally adopted. We could've had a completely different set of axioms and completely different mathematical concepts as a result. The reason why ZFC is chosen is because it results in concepts that are applicable in the real world. If a new independent axiom could give a rise to some useful concept applicable across many different mathematical fields, eventually ZFC+X might be adopted as the universal axiom system and it wouldn't be necessary to write "assuming X" in every proof.
> other axioms are considered so self-evident that it's not even worth mentioning.
Maybe "considered so common" would be a more precise wording.
I'd also argue that the design space within finite math is reasonably limited. I have never seen an interesting alternative axiomization. For infinite math there is of course much more room.
Maybe "considered so common" would be a more precise wording.
I'd also argue that the design space within finite math is reasonably limited. I have never seen an interesting alternative axiomization. For infinite math there is of course much more room.
Yes, you're right, this entire fight is only about which avenues of research are fashionable. For the vast majority of mathematicians, this does not affect their work, and for the mathematicians whose work is directly affected, "answering" this question is like biologists answering the question, "Lions or tigers?" Some people's research will become more fashionable and other people's research will become less fashionable. No answer to this question will be of any more consequence than the philosophical conclusions that a number cannot represent zero quantity, negative quantities are nonsensical, parallel lines never meet, etc.
“Woodin showed that if you can just reach the level of the supercompacts, then there’s an overflow and your inner model picks up all the bigger large cardinals as well,” Koellner explained. “That was a sort of landscape shift. It provided this new hope that this approach can work. All you have to do is hit one supercompact and then you’ve got it all.”
This is the mathematical equivalent of "640K ought to be enough for anybody." Somebody's going to want something more, or something different, and they're going to do what they want. In the long run, mathematicians only care what assumptions generate neat and/or useful mathematics. In the short run this might affect young researchers' employment prospects and might influence them to choose different problems for that reason.
“Woodin showed that if you can just reach the level of the supercompacts, then there’s an overflow and your inner model picks up all the bigger large cardinals as well,” Koellner explained. “That was a sort of landscape shift. It provided this new hope that this approach can work. All you have to do is hit one supercompact and then you’ve got it all.”
This is the mathematical equivalent of "640K ought to be enough for anybody." Somebody's going to want something more, or something different, and they're going to do what they want. In the long run, mathematicians only care what assumptions generate neat and/or useful mathematics. In the short run this might affect young researchers' employment prospects and might influence them to choose different problems for that reason.
> Its not like the real world places any restrictions on math
There has been an ongoing debate about whether that statement is true for the last couple of millennia. Many people are arguing that mathematics were not developed out of the blue, but to describe the physical world around us.
There has been an ongoing debate about whether that statement is true for the last couple of millennia. Many people are arguing that mathematics were not developed out of the blue, but to describe the physical world around us.
I would agree that math an the real world are related, but not that the physical world should influence math. If you discover an interesting and self consistent system, than it seems that eventually there is a good chance you will find a connection with physics. But you don't need to let perceptions about the physical world influence the systems you look for.
> and thats totally fine because the real world does not contain infinities
Anyone who positively claims that infinities do or do not exist in nature don’t know what they’re talking about.
Physics is full of infinities. They show up all the time in relativity (gravitational singularities) and quantum physics (renormalization, rigged Hilbert spaces with integration over infinite distributions).
For some of these infinities (like the ones we have to deal with in quantum mechanics), it’s unclear (to me, at least) whether or not they have any sort of “real” equivalent. For things like gravitational singularities, our current theories predict the physical existence of infinite quantities (density, gravitational field, etc.).
Anyone who positively claims that infinities do or do not exist in nature don’t know what they’re talking about.
Physics is full of infinities. They show up all the time in relativity (gravitational singularities) and quantum physics (renormalization, rigged Hilbert spaces with integration over infinite distributions).
For some of these infinities (like the ones we have to deal with in quantum mechanics), it’s unclear (to me, at least) whether or not they have any sort of “real” equivalent. For things like gravitational singularities, our current theories predict the physical existence of infinite quantities (density, gravitational field, etc.).
But it's also very obvious that general relativity can only be an effective theory that fails for very strong fields. No one I know actually thinks there are singularities, those just point to places where GR starts to significantly deviate from reality. Navier–Stokes equations are very accurate on macroscopic scale, but regardless of their mathematical properties, there are no actual blow ups in real world, because ultimately, the liquid is composed from atoms. Likewise, the fields in GR must be ultimately quantized in some way, which will almost certainly break the singularities.
The infinities in QFT are really just a hack. I mean, all those calculations are perturbational in the first place, so they can be hardly considered a "true" picture of reality.
The infinities in QFT are really just a hack. I mean, all those calculations are perturbational in the first place, so they can be hardly considered a "true" picture of reality.
Precisely stating your assumptions happens in a limited way in everyday mathematics, enough so that the project of finding the minimal set of assumptions for mathematical theories is a niche project: https://en.wikipedia.org/wiki/Reverse_mathematics. By everyday mathematics, I mean mathematics outside of set theory/logic/ foundations of mathematics.
Rather, what everyday mathematicians do is have a set of axioms that are "safe", and do not need discussion, along with other axioms that require explicit note. They then prove things relative to those axioms, make a limited effort to only use the "unsafe" axioms, and make no effort to restrict their use of safe axioms.
Mathematicians will typically describe their philosophy of mathematics as formalist, but it's arguable that everyday mathematicians act much more like Platonists with respect to ZFC.
Rather, what everyday mathematicians do is have a set of axioms that are "safe", and do not need discussion, along with other axioms that require explicit note. They then prove things relative to those axioms, make a limited effort to only use the "unsafe" axioms, and make no effort to restrict their use of safe axioms.
Mathematicians will typically describe their philosophy of mathematics as formalist, but it's arguable that everyday mathematicians act much more like Platonists with respect to ZFC.
Amen. Not a mathematician, but this perfectly articulates my thoughts on the subject.
>the real world does not contain infinities
I have no idea what "real world" even means, but I'd say that infinity does not exist within anything that can be conceptualized.
>the real world does not contain infinities
I have no idea what "real world" even means, but I'd say that infinity does not exist within anything that can be conceptualized.
> infinity does not exist within anything that can be conceptualized.
Yet math is consistently being conceptualized?
Enabling clear thinking about concepts outside of the real world is the while point of math.
Yet math is consistently being conceptualized?
Enabling clear thinking about concepts outside of the real world is the while point of math.
>Yet math is consistently being conceptualized?
Sure, how else could we capture it or convey it to others? The thing is that I don't see it as that different than conceptualizing things in the "real world". In the latter, conceptualization is done on input of the senses, while in math conceptualization is done on inputs from a higher level of abstraction. In both cases, something is defined and then manipulated for something useful. It's just that I believe that whatever is defined automatically becomes a finite concept otherwise it could not be grasped and manipulated, and a finite infinity seems problematic...
Sure, how else could we capture it or convey it to others? The thing is that I don't see it as that different than conceptualizing things in the "real world". In the latter, conceptualization is done on input of the senses, while in math conceptualization is done on inputs from a higher level of abstraction. In both cases, something is defined and then manipulated for something useful. It's just that I believe that whatever is defined automatically becomes a finite concept otherwise it could not be grasped and manipulated, and a finite infinity seems problematic...
> math is consistently being conceptualized
What does this mean?
What does this mean?
Quantum mechanics has several facets which correspond to path integrals, infinite-dimensional operators, and infinite series, sums, and products.
True - you don't even need to venture into quantum mechanics, the concept of infinity has been useful in practical science far before anyone dreamed of quantum mechanics - but there's a difference between a definition of what happens to a formula when a limit approaches infinity and claiming to somehow understand infinity as a concept. I mean when you deal with infinite-dimensional operators, infinite-series etc. does it matter what ordinal of infinity is used?
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When your hand moves, does it pass through an infinite or a finite number of positions at a finite or infinite points in time? Infinity need not be big.
If you are going to play with the "real world", you are going to have to deal with continuous math; i.e. the real numbers at least. At which point you get to deal with infinity somehow.
I believe that potential infinity (i.e. the idea of intervals becoming ever smaller and smaller) will be sufficient to deal with the "real world". But the infinities described in the article are of a different kind.
I don't think there will ever be a situation like: "According to the axiom of choice, this photon should go to the left, but according to the axiom of something else, this photon should go to the right." Or: "If there is such a cardinal X that mumbo-jumbo(X) is true, then the photon should go to the left, otherwise it should go to the right."
Therefore, this part of math is just playing with words. An interesting game, especially if you can get paid for it, and it can provide indirect benefits such as greater insight into how to correctly play with words, but ultimately just a game with no correspondence to the "real world".
I don't think there will ever be a situation like: "According to the axiom of choice, this photon should go to the left, but according to the axiom of something else, this photon should go to the right." Or: "If there is such a cardinal X that mumbo-jumbo(X) is true, then the photon should go to the left, otherwise it should go to the right."
Therefore, this part of math is just playing with words. An interesting game, especially if you can get paid for it, and it can provide indirect benefits such as greater insight into how to correctly play with words, but ultimately just a game with no correspondence to the "real world".
If infinity is being boundless then to make anything infinity have the bounds as unknown. A practical use is in Haskell's lazy evaluation. Although maybe I'm missing the point completely.
“What truly infinite objects exist in the real world?”
I'm not sure what "infinite object" means, but I do find it interesting that we can do the math anyway using only a finite number of symbols.
I'm not sure what "infinite object" means, but I do find it interesting that we can do the math anyway using only a finite number of symbols.
My dad would answer when I was a kid, at the end, there is just a wall, but then I’d ask what what’s beyond the wall and that was the end of the conversation
Title needs a (2013) at the end.
In my opinion, trying to settle the matter by adopting a new axiom is a bit like proving that blue is the best color for cars by requiring all cars to be painted blue.
In my opinion, trying to settle the matter by adopting a new axiom is a bit like proving that blue is the best color for cars by requiring all cars to be painted blue.
A different viewpoint: "adding a new axiom" is synonymous with "further restricting the things under discussion".
Imagine biologists decide that "DAG" is the correct abstraction for parent-child relationships. Then they spend years trying to prove the obviously-true fact that parents can't have infinitely many children. Of course, they never can, because DAG nodes can have infinitely many children. Finally they realize they chose the wrong abstraction, they should've chosen "DAGs where all nodes have finitely many children".
To suggest adding a new axiom to ZFC is to suggest that ZFC was an over-broad abstraction for capturing what things are supposed to be "sets".
Imagine biologists decide that "DAG" is the correct abstraction for parent-child relationships. Then they spend years trying to prove the obviously-true fact that parents can't have infinitely many children. Of course, they never can, because DAG nodes can have infinitely many children. Finally they realize they chose the wrong abstraction, they should've chosen "DAGs where all nodes have finitely many children".
To suggest adding a new axiom to ZFC is to suggest that ZFC was an over-broad abstraction for capturing what things are supposed to be "sets".
This sort of thing just strikes me as so pointless. How could one possibly decide? Obviously, if we're talking about new axioms that aren't provable from the old ones, you can't use those old ones to settle the matter. The axioms under discussion don't really affect normal math so you can't appeal to their usefulness there (unlike with, say, the axiom of choice, which famously was controversial for a while). And I don't think people can really have much of a claim to have any good intuition for such things, or else we'd have had such an axiom a long time ago. My bet is neither of these will become widely accepted.
That said, if I had to bet on one of the two becoming accepted, I'd go with "V = ultimate L". Why? Because mathematics already has an axiom roughly analogous, that doesn't actually affect normal mathematics but prunes away sets we don't need so people don't waste their time with them. I'm talking, of course, about the axiom of foundation. It's like, why is that even in ZFC? Ordinary math doesn't actually care about the material content of sets; it really affects nothing. The only thing it does, is that, well, without it, non-well-founded sets are something of a problem, in that they could exist, or they could not, and you can't really prove much about them either way. There are non-well-founded set theories that replace the axiom of foundation with an axiom of antifoundation, allowing you to actually prove things about non-well-founded sets -- Peter Aczel wrote a good book on the subject -- but if you just take it out and don't put anything in to replace it, then, well, you just can't really say anything about the matter. The axiom of foundation prunes these irrelevant sets away and tells you to focus on actual math rather than this mystery that's not really much of a mystery at all. So I'd bet on another axiom along those lines becoming accepted, rather than one that does the opposite. After all, if we want to take the opposite perspective that anything that can exist ought to... well, then why not non-well-founded sets, too?
(Note: Not a logician, my understanding of the two axioms being discussed is based pretty much entirely on this article, so the analogy I claim may not actually work as well as I think it does.)
That said, if I had to bet on one of the two becoming accepted, I'd go with "V = ultimate L". Why? Because mathematics already has an axiom roughly analogous, that doesn't actually affect normal mathematics but prunes away sets we don't need so people don't waste their time with them. I'm talking, of course, about the axiom of foundation. It's like, why is that even in ZFC? Ordinary math doesn't actually care about the material content of sets; it really affects nothing. The only thing it does, is that, well, without it, non-well-founded sets are something of a problem, in that they could exist, or they could not, and you can't really prove much about them either way. There are non-well-founded set theories that replace the axiom of foundation with an axiom of antifoundation, allowing you to actually prove things about non-well-founded sets -- Peter Aczel wrote a good book on the subject -- but if you just take it out and don't put anything in to replace it, then, well, you just can't really say anything about the matter. The axiom of foundation prunes these irrelevant sets away and tells you to focus on actual math rather than this mystery that's not really much of a mystery at all. So I'd bet on another axiom along those lines becoming accepted, rather than one that does the opposite. After all, if we want to take the opposite perspective that anything that can exist ought to... well, then why not non-well-founded sets, too?
(Note: Not a logician, my understanding of the two axioms being discussed is based pretty much entirely on this article, so the analogy I claim may not actually work as well as I think it does.)
They speak of the Philsophy of Mathematics. I typed a lot out and chose to delete it. In short, it is considered the highest order among Mathematicians. However, us Applied Mathematicians just get on with the real work and have pretty much figured out that they are insane.
It's a bit like true random and infinity. It's best not to think about it to much and only useful for post doc bar conversations or impressing stoned Econ undergrads of your preferred gender. Even then, it's best not to think about it too much.
It's a bit like true random and infinity. It's best not to think about it to much and only useful for post doc bar conversations or impressing stoned Econ undergrads of your preferred gender. Even then, it's best not to think about it too much.
> They speak of the Philsophy of Mathematics. In short, it is considered the highest order among Mathematicians.
Speaking as a pure mathematician: It's really not.
When I say this stuff is pointless, that it doesn't affect ordinary mathematics, I'm not just talking applied math; I'm talking, well, all of ordinary mathematics, pretty much. Number theory, combinatorics... every now and then a question comes up whose answer depends on the continuum hypothesis admittedly but it's seriously rare (and on those few occasions when it has happened, I think often someone comes along and says that this means that you were really asking the wrong question in the first palce).
I mean, OK, there are people such as Harvey Friedman who work out arithmetic consequences of such assumptions. But the statements they come up with are seriously contrived, not the sort of thing ordinary mathematicians care about, even if the statements no longer involve infinities.
Hell I work with infinities sometimes and I don't care about this stuff! Of course, I'm not a proper logician or set theorist. But doing computations with ordinals and well partial orders is fun... and the answers you get never end up depending on this sort of stuff. :P
Speaking as a pure mathematician: It's really not.
When I say this stuff is pointless, that it doesn't affect ordinary mathematics, I'm not just talking applied math; I'm talking, well, all of ordinary mathematics, pretty much. Number theory, combinatorics... every now and then a question comes up whose answer depends on the continuum hypothesis admittedly but it's seriously rare (and on those few occasions when it has happened, I think often someone comes along and says that this means that you were really asking the wrong question in the first palce).
I mean, OK, there are people such as Harvey Friedman who work out arithmetic consequences of such assumptions. But the statements they come up with are seriously contrived, not the sort of thing ordinary mathematicians care about, even if the statements no longer involve infinities.
Hell I work with infinities sometimes and I don't care about this stuff! Of course, I'm not a proper logician or set theorist. But doing computations with ordinals and well partial orders is fun... and the answers you get never end up depending on this sort of stuff. :P
Well, yes. They are insane.
Infinity is a rubbish idea, by the way. I've given it much thought and concluded it is best to just not think about it. Cantor was a lunatic, and I'm not entirely certain that infinity isn't a causal factor.
Don't get me started. It doesn't end well. ;-)
Eventually I go off on the absurdity of 'true random' and our presumption to know the differences between it and unpredictability. It's hubris and madness, all the way down.
They really should stop teaching mathematics by rote. If you're familiar with my posting history, I very seldom say 'should.'
Infinity is a rubbish idea, by the way. I've given it much thought and concluded it is best to just not think about it. Cantor was a lunatic, and I'm not entirely certain that infinity isn't a causal factor.
Don't get me started. It doesn't end well. ;-)
Eventually I go off on the absurdity of 'true random' and our presumption to know the differences between it and unpredictability. It's hubris and madness, all the way down.
They really should stop teaching mathematics by rote. If you're familiar with my posting history, I very seldom say 'should.'
Eh, everyone but the set theorists ignores the set theorists at this point. I think some exciting things happened in the 60's with forcing, but a lot of the subsequent developments seem pretty contrived.
Non-well-founded sets seem to be grammars from a CS point of view. As soon as you admit the Kleene-star '*' operator you have potential infinity. If you allow circular grammar definitions, then you are specifying non-well-founded-sets.
When parsing finite structures, it just so happens that the parsing terminates, but it's easy to extend to open-ended (potentially infinite) streams. There are even type system constructs and language expressions for them, but only executable in lazy languages, or systems open to external input.
CS has also made theoretical progress, perhaps the most surprising is differentiating regular expressions and other regular datatypes:
http://en.wikipedia.org/wiki/Brzozowski_derivative
http://blog.sigfpe.com/2005/05/derivatives-of-regular-expres...
http://strictlypositive.org/diff.pdf
I also recommend 'The Liar' by Barwise & Etchemendy.
When parsing finite structures, it just so happens that the parsing terminates, but it's easy to extend to open-ended (potentially infinite) streams. There are even type system constructs and language expressions for them, but only executable in lazy languages, or systems open to external input.
CS has also made theoretical progress, perhaps the most surprising is differentiating regular expressions and other regular datatypes:
http://en.wikipedia.org/wiki/Brzozowski_derivative
http://blog.sigfpe.com/2005/05/derivatives-of-regular-expres...
http://strictlypositive.org/diff.pdf
I also recommend 'The Liar' by Barwise & Etchemendy.
Higher mathematics is pointless and had always been pointless. People do it because it's interesting and fun.
Topology is a great example of higher mathematics with important connections to many applied endeavors:
https://en.wikipedia.org/wiki/Topology#Applications
https://math.stackexchange.com/questions/73690/real-life-app...
https://en.wikipedia.org/wiki/Topology#Applications
https://math.stackexchange.com/questions/73690/real-life-app...
This feels like the mathematical equivalent of string theory, an untestable and apparently unprovable 'set' of ideas has shown up and captured the minds of mathematicians. The impact of these questions is that it seems to block progress?
The reason to pay attention to this stuff is that it might (just might) affect other conjectures in mathematics.
A very clear example here is the continuum hypothesis. It depends on which axiom you choose whether or not the hypothesis holds. If other hypotheses (e.g. Goldbach, Riemann) turn out to have different results in differing axiom systems, that tells us something about those conjectures.
Specifically, it tells us that maybe the conjectures might not be that important.
A very clear example here is the continuum hypothesis. It depends on which axiom you choose whether or not the hypothesis holds. If other hypotheses (e.g. Goldbach, Riemann) turn out to have different results in differing axiom systems, that tells us something about those conjectures.
Specifically, it tells us that maybe the conjectures might not be that important.
> If other hypotheses (e.g. Goldbach, Riemann) turn out to have different results in differing axiom systems, that tells us something about those conjectures.
To be clear, you shouldn't (can't ?) have something like that Riemann is true under ZFC and false under ZFC + Continuity, because that would imply that ZFC + Continuity is inconsistent. You could have that Riemann is true under ZFC + Continuity but unprovable under ZFC, though.
To be clear, you shouldn't (can't ?) have something like that Riemann is true under ZFC and false under ZFC + Continuity, because that would imply that ZFC + Continuity is inconsistent. You could have that Riemann is true under ZFC + Continuity but unprovable under ZFC, though.
Indeed, if we presume ZFC + Continuity to be consistent, then your situation cannot happen.
However, it could be that ZFC + Continuity has Goldbach true, and ZFC + Forcing has goldbach false.
I should say I recall reading that if the Riemand hypothesis is false, then it is provably so in ZFC, hence my inclusion of the Goldbach conjecture. [1, 2]
[1] https://mathoverflow.net/questions/79685/can-the-riemann-hyp...
[2] https://www.youtube.com/watch?v=O4ndIDcDSGc&ab_channel=Numbe...
I should say I recall reading that if the Riemand hypothesis is false, then it is provably so in ZFC, hence my inclusion of the Goldbach conjecture. [1, 2]
[1] https://mathoverflow.net/questions/79685/can-the-riemann-hyp...
[2] https://www.youtube.com/watch?v=O4ndIDcDSGc&ab_channel=Numbe...
The main difference: this is not a new "law of mathematics" so much as a new law of set theory. No one but set theorists will be affected, and no other mathematicians care much.
Foundations are not nearly as set as undergrad math teaches you. I don't mean the Axiom of Choice (which pretty much no one has any problems with anymore). However, various large cardinal axioms are pretty standard in advanced Algebraic Geometry (for example, the existence of Grothendiek Universes). Set theorists work with all sorts of large cardinal axioms all the time, and for a while it was fashionable to assume various generalized continuum hypotheses. Graph theorists, for some applications, assume "V=L", because that makes things tidy. It's fine, and no one really has a problem with it, nor do these disagreements really cause problems, even though some of these axioms are contradictory. (And even when various large cardinal axioms are assumed, often people work with an intuitive ill-founded set theory with the assumption that it can be made well-founded by ramifying using the large cardinals instead.)
The 21-century understanding is something like: there's no need for there to be "ultimate" foundations. As mentioned, there is little impact on "ordinary" mathematics from different axioms of set theory, so maybe the set theorists should choose axioms of set theory, but that needn't affect other mathematicians. Obviously someone should study set theory, it's cool and there are interesting theorems and so on; and if they find that "V = ultimate L" is a useful axiom for their field of study, nothing wrong with that, much like commutative algebra is a fine field of study though of course non-commutative algebra is interesting too. But there's no realistic sense in which group theory is "defined in terms of" set theory—you can tell, because group theorists tend not to care about set theoretic axioms, and have no dog in choosing between "V=ultimate L" and forcing.
The argument between "V = ultimate L" and the forcing axioms are a debate over what "set theory" should study, and each mathematician involved thinks that it should study the sort of things that they study—whether that means forcing or large cardinal axioms. Large cardinal axiom people have a well-developed kind of work they want to do: defining various forms of large cardinals, proving implications between them, and then discovering that all large cardinal axioms are totally ordered by proof strength. It's good, interesting work, and "V = ultimate-L" will really complete the research program they've been pursuing; or, proving independence results for various theorems, and a forcing axiom would really put the machinery they work with deep into the foundations of set theory, making that work much simpler and at the same time deeper. That, too, is a valuable field of research, in fact it has led to the only Fields medal in logic.
I assure you that your personal understanding of sets will not be affected much by which axiom which mathematicians choose to adopt.