Why Do Certain Large Cardinals Have Specific Properties?

[tzimie]

Reading this: https://en.wikipedia.org/wiki/List_of_large_cardinal_properties

I perfectly understand the motivation for the inaccessible cardinal, it looks very natural from the theory of ordinals, but everything beyond makes no sense to me. Why Mahlo? Why superstrong? It looks like a game "What if there is so large cardinal that it has the following weird property: <you name it>"

While "what-if" game is good for math, I fail to understand what was driving the choice of the specific weird properties in the "formula" (I stressed in italic) above. Especially reading this: https://en.wikipedia.org/wiki/Vopěnka's_principle

Vopěnka's principle was originally intended as a joke: Vopěnka was apparently unenthusiastic about large cardinals and introduced his principle as a bogus large cardinal property, planning to show later that it was not consistent

 

[chiro]

Hey tzimie

Current mathematics is based on an understanding of one-dimensional quantities and they have order (i.e. can be used with equality, greater or less than and combinations thereof).

The point of cardinality is to make some sense of spaces (with sets) so that they can be compared.

I do agree it is often ridiculous to just come up with classifications that serve not much of a use bit it's important in current mathematics so as to know how much information is in a space and how the different spaces relate to each other in terms of subsets and the cardinalities they have.

Because they are fundamental components/measures of sets and because sets are fundamental to mathematics they become fundamental in and of itself.

 

[Zafa Pi]

Reading this: https://en.wikipedia.org/wiki/List_of_large_cardinal_properties

I perfectly understand the motivation for the inaccessible cardinal, it looks very natural from the theory of ordinals, but everything beyond makes no sense to me. Why Mahlo? Why superstrong? It looks like a game "What if there is so large cardinal that it has the following weird property: <you name it>"

While "what-if" game is good for math, I fail to understand what was driving the choice of the specific weird properties in the "formula" (I stressed in italic) above. Especially reading this: https://en.wikipedia.org/wiki/Vopěnka's_principle

This can happen in other fields as well. People get attached to a field and want to publish and this is what often happens.

 

[nomadreid]

There is a lot of lack of respect for the field of Model Theory, specifically Large Cardinals. The "weird properties" tend not to be so weird when you get into them. Yes, you can do mathematics without knowing what is justifying the steps you are doing, but I think any mathematician would rebel at that. Why Mahlo? If you are dealing with a class of continuous increasing functions, and you want to make sure that you get them to converge, then you might needwant to assume that this is (relatively) consistent, which you can do by proving that the existence of a Mahlo cardinal is equiconsistent with the standard model of arithmetic. Or if you want to prove properties of Boolean Algebras using filters, the existence of Mahlo cardinals will give you the green light. For Superstrong and other cardinals that deal with Reflection Principles, you will find a good discussion here: logic.harvard.edu/koellner/ORP_final.pdf (He discusses Vopenski as well.) Perhaps the nicest example is measurable cardinals. If you want to deal with a physical universe where there are principles which are true except at insignificant points (or, to put it another way, if you want to be able to integrate over discontinuous spaces), then you might want to adopt the ultraproduct construction... but how do you know that this construction is valid? You prove that the existence of a measurable cardinal is equiconsistent etc. Some of the results melt into recursion theory, such as the one that the existence of a Ramsey cardinal proves that there are non-constructible real numbers. And so forth. True, the large cardinals did not come up with a model that would be as acceptable as ZFC and allow one to either accept or reject the Continuum Hypothesis, let alone the Generalized Continuum Hypothesis, but that doesn't make it worthless. I think you could look at any field of advanced mathematics and point to concepts that appear weird at first glance.

 

[tzimie]

There is a lot of lack of respect for the field of Model Theory, specifically Large Cardinals.

Sorry for breaking your heart )))
But I am not mathematician at all - but the subject is just so interesting so I can't stop reading about it. So it is not so bad after all.

you will find a good discussion here: logic.harvard.edu/koellner/ORP_final.pdf (He discusses Vopenski as well.)

Thank you for the link and for the highlighting of some points.

After reading more about that stuff, I had also found that there is some parallelism with large countable ordinals, so if ZFC proves the existence of Mahlo ordinal, it makes sense to think about the Mahlo cardinal (even it is not decidable in ZFC).

 

[tzimie]

If you want to deal with a physical universe...

But what is the "strength" of physics as mathematical theory? (giving the word "strength" a precise meaning, like epsilon-naught is an ordinal which measures the strength of PA). I am afraid physics is very weak because it is finite. It comes from the finite Hubble volume and finite amount of information which can be stored in a given volume (Bekenstein bound). So looks like physics is not only countable but is also finite, and in that sense it is much weaker, than, say, Conway's "Game of Life" universe - where, as you know, a REAL infinite turing machine can be constructed. In our real universe it is not possible.

... unless there is infinite informational complexity hidden in real/complex variables everywhere. However, as in physics functions tend to be continuous, they can be defined specifying their values in rational points only, which makes our universe countable (even infinite in this case).

 

[nomadreid]

... physics is very weak because it is finite. It comes from the finite Hubble volume and finite amount of information which can be stored in a given volume (Bekenstein bound). So looks like physics is not only countable but is also finite, ...

Hm, here we verge on the old debate as to whether the wave function is "real". So, although the precision of each eigenvalue may be restricted to the rationals, the number of possible eigenvalues may be uncountable. In any case, Hilbert space is definitely infinite, and I would say that Hilbert space belongs to physics.

However, as in physics functions tend to be continuous, they can be defined specifying their values in rational points only, which makes our universe countable (even infinite in this case).

If I understand this last point correctly, you are saying that since every real number can be defined as the limit of a sequence of rational numbers, we don't need the reals. But if you throw out the reals, and only keep the sequences without assuming closure, you have a universe (or a theory thereof) which is not closed. This tends to be unsatisfactory. Also, yes every real can be so defined (although one has to be careful with the word "defined", because in the set-theoretical sense, there are reals which are not definable in the standard model), but there are still an uncountable number of such sequences. Oh, and non-standard analysis is so much nicer than the messy limits anyway.

 

[tzimie]

Hm, here we verge on the old debate as to whether the wave function is "real".

Well, I have found an answer here: https://arxiv.org/abs/0704.0646

Also, yes every real can be so defined (although one has to be careful with the word "defined", because in the set-theoretical sense, there are reals which are not definable in the standard model), but there are still an uncountable number of such sequences. Oh, and non-standard analysis is so much nicer than the messy limits anyway.

You are right, but is it possible that there are physically measurable observables which values depend on (i) undecidable statements in TOE or (ii) of (non constructive) existence of objects like Vitali sets usually given by AC? I doubt it (it is a conjecture). Anyway, it leads to very interesting consequences:
1. TOE is too weak (or finite) so there are neither undecidable statements nor effect of assuming/denying AC/CH//Large cardinals
2. TOE has such statements ALWAYS in a controllable manner so they never affect the observables. This is actually a very strong statement
Finally, if this conjecture is false:
3. You can actually measure if, say, CH is true in our Universe (which is a big WOW)

 

[nomadreid]

Well, I have found an answer here: https://arxiv.org/abs/0704.0646

Max Tegmark is always a joy to read. (He presents his "Physics is a subset of mathematics" thesis as a bold if reasonable one: but it is tame compared to Gödel's notebooks outlining his Platonism. Off-the-wall, so not usually referred to.) Of course you can find huge numbers of articles arguing both sides, but here is a less philosophical one: https://[B]arxiv[/B].org/pdf/1412.6213

is it possible that there are physically measurable observables which values depend on (i) undecidable statements in TOE or (ii) of (non constructive) existence of objects like Vitali sets usually given by AC? I doubt it (it is a conjecture). Anyway, it leads to very interesting consequences:
1. TOE is too weak (or finite) so there are neither undecidable statements nor effect of assuming/denying AC/CH//Large cardinals
2. TOE has such statements ALWAYS in a controllable manner so they never affect the observables. This is actually a very strong statement
Finally, if this conjecture is false:
3. You can actually measure if, say, CH is true in our Universe (which is a big WOW)

Although it is a truism that, given that we do not know what the TOE will look like and therefore it is not possible to answer such questions as to whether undecidable statements will play a role, I think it is safe to assume that the Vitali sets will only play the indirect role that they were meant to, to wit, to show what sort of assumptions are needed to get Lebesgue integrals where you want them. In a way, without mixing too strongly uncertainty and undecidability, it is likely that in TOE the probabilistic aspect of quantum physics will be retained, so that the theory will be considered complete only if one restricts the theory to referring only to probabilities; once you allow it to talk about specific measurements, you get an incomplete theory -- which is not bad, but it does indicate the presence of undecidable statements. But more to the point: referring to the existence of large cardinal axioms as "undecidable" axioms is assuming ZFC (all of whose axioms are undecidable with respect to the trivial theory), and once you have ZF, you have undecidable statement lurking around, such as its consistency, which you implicitly assume in order to apply standard mathematics. In a word, you aren't going to get away from undecidable statements, even if you stick to arithmetic. And I don't know of many physicists who would willingly dispense with arithmetic.
As far as CH, I would be interested to see how one could directly measure anything that could tell whether it is true or not. Not that anyone would want to: it has proven to be pretty useless beyond the original points it was invented for.

 

[tzimie]

Just found something related to physics and undecidable statements:
https://en.wikipedia.org/wiki/Hypercomputation

A real computer (a sort of idealized analog computer) can perform hypercomputation[4] if physics admits general real variables (not just computable reals), and these are in some way "harnessable" for useful (rather than random) computation. This might require quite bizarre laws of physics (for example, a measurable physical constant with an oracular value, such as Chaitin's constant), and would require the ability to measure the real-valued physical value to arbitrary precision.

• Similarly, an neural net that somehow had Chaitin's constant exactly embedded in its weight function would be able to solve the halting problem,[5] though constructing such an infinitely precise neural net, even if you somehow know Chaitin's constant beforehand, is impossible under the laws of quantum mechanics