The $\omega_1$ discussion 1: Physical meaning of a mathematical object with cardinality $\omega_1$

Secret

Nov 29, 2017 15:49

Btw, regarding ZFC, I was rambling in the Rambles recently as I tried to figure out how actual infinity and all those crazy infinite notions in ZF arises from considering potential infinity (and try to temporary purge myself from ZF knowledge to pretend I don't know anything except basic counting) and more importantly, to figure what is a minimalistic notion of infinities, and I realise I got stuck very easily

If one thing, infinity being so weird in modern mathematics seemed to heavily tied to the way ZF and set theory in general is structured. I am having more suspicion that perhaps a lo…

user21820

@Secret That is in fact true. A lot of 'facts' about infinity are actually artefacts of the foundational system.

Secret

Currently in my investigation, only potential infinity can be motivated, by considering the philosophical notion of something that goes forever.

But actual infinity is in some sense quite ad hoc

there is basically no reason in real life application for us to imagine something larger than anything we can access. Like why will we be interested in a notion of inaccessibility (that seemed to be the minimal requirement for something to be any of the large infinities)

user21820

@Secret That's why it is an artefact of the foundational system. Even ω[1] does not seem to make sense from a constructive point of view.

No need to go so far as inaccessibles.

ω[1] as a classical set, I mean.

Or as a von Neumann ordinal.

Countable ordinals as a type makes sense, in the same way natural numbers as a type makes sense. Namely whatever you can prove is a countable ordinal gets accepted into that type. Nothing ever gets rejected once it is accepted.

Secret

Well, leaky and I pretty much had a preliminary conclusion, that $\omega_1$ is already inaccessible in a sense no countable fundamental sequences or repetitions of it in any number of iterations below $\aleph_1$ itself can access it.

I am suspecting the thing that really characterise the notion of uncountability is "inaccessible from countably many steps" and in general a successor cardinal is basically "inaccessible from predecessor cardinal many steps"

Indeed, some of the rambles I made earlier (which I later deleted as Mathworks need major cleanup due to its unexpected gain in purpose)…

And in type theory, it is much simpler as you said, all countable ordinals are of the same type, done, nothing ambigurous about it

user21820

@Secret Well I don't know whether I linked you to the post I made on replacement:

1

A: Why is replacement true in the intuitive hierarchy of sets?

Suppose that as each stage $S$ is completed, we take each $y$ in $x$ which is formed at $S$ and complete the stage $S_y$. When we reach the stage at which $x$ is formed, we will have formed each $y$ in $x$ and hence completed each stage $S_y$ in $\mathbf S$. This is actually a circular justi...

My answer is actually the correct one, but as you can see people are afraid to face the truth. At least the logician Boolos was forthright about it.

Secret

Nov 29, 2017 16:00

That's a new post indeed. I am reading it now. Meanwhile I had some comments about asaf's:

But if you don't have $V_{\omega_1}$, how can you prove Borel determinacy? :( — Asaf Karagila Oct 18 '16 at 11:38

user21820

You can see that even granting the powerset and union capabilities, you can't get an uncountable number of stages, and so can't construct the stage in the cumulative hierarchy containing ω[1], without already having ω[1] to furnish you all the labels you need for your stages...

Boolos confirms my initial guess (in my original post before I found his remark) that V[ω[1]] forms a model of the iterative conception of sets.

Note that you still can construct the set of all countable well-orderings on N.

But you can't extract the canonical ordinal from this.

Secret

(cont.)
It is true we now knew how ad hoc $\omega_1$ is. The problem is what should we do to fix borel algebra since that thing defines measure theory and it turns out borel algebra is a set $G^{\omega_1}$

$\omega_1$ also appear in an important topological counterexample the long line. The nonexistence of $\omega_1$ means we need to figure out how to retain these important cases

Basically on those few days ago when I realise $\omega_1$ is so arbitrary, it seems a lot of things from measure theory will fall apart, and then question is then how to prevent is since measures are so important …

user21820

@Secret ω[1] is not important in the part of measure theory that is needed in the real world.

Are you surprised?

I know about the long line too, and it's true that ω[1] can be used in ZFC to do all sorts of interesting things. But at the end of the day I don't see any practical use.

47

Q: Physical meaning of the Lebesgue measure

Question (informal) Is there an empirically verifiable scientific experiment that can empirically confirm that the Lebesgue measure has physical meaning beyond what can be obtained using just the Jordan measure? Specifically, is there a Jordan non-measurable but Lebesgue-measurable subset of ...

Secret

@user21820 I think I am quite surprised, cause I do see it pop up in quite a lot of real analysis textbook where various closure properties of spaces were discussed

user21820

@Secret I'm quite sure most of them are about counter-examples to something.

@LeakyNun: By the way, if you read my post on replacement and find it okay, you could help to downvote the silly circular answer by the asker. It can't be removed by deletion.

Secret

Nov 29, 2017 16:13

The way I handle replacement nowadays is to ensure every function I use (unless I am explicitly using the axiom of choice for certain applications like Baire category theorem) can be wrote as a finite string of instructions so that the function becomes an algorithm that if plug into a computer, should compute its image as required

so perhaps I might be using the constructive or possibly predicative portion of replacement

user21820

Your programs include oracles?

If not even the halting problem is not accessible to you, and you don't even need any set theory; just PA will do.

Secret

No unless I need some important uncomputable reals

Yeah I think PA with its induction scheme will compute the image of those functions for me, I think that's how I think of them since I tend to treat functions as a list where inputs feed in and spits out a list

user21820

Well if you have finite Turing jumps, and consider every object as some description, then you can have replacement for functions on N, and you still won't get any weird things like ω[1].

Because countable stuff are closed under countable stuff anyway.

Secret

Indeed, and one of my rambles I deleted discussed that in a finite turing jump model, the collection of all ordinals will be $[0,\omega_{n+1}^{CK})$ which is powerful enough to do most analysis

user21820

Yes as I said earlier I'm sure that all applicable real analysis can be done in higher-order arithmetic (HOA), and virtually all in just ACA.

ACA has all finite jumps.

Secret

Nov 29, 2017 16:30

Another footnote about set theory: Even $\omega$ will become unreachable if axiom of infinity is not part of the default axioms of ZF (thus in such hypothetical scenario, one will need to introduce it as some models of ZF)

So in general, my conjecture is that actual infinities does not exists before they are being defined by axioms

→ 4 messages moved to Mathematics

@AsafKaragila: Yea but some people want AC, and hence can't have AD, even if AD is the natural generalized version of determinacy. So for them, my question is how they determine which determinacy is the 'right' one. In a sense, $ω_1$ and $ω_1^{CK}$ are like large cardinals, each in their own sense, the first from the constructive perspective and the second from the computability perspective. And just a while ago I was going to suggest some stronger iterative principle that might get us up to $V_{\beth_{ω_1}}$ but no further... — user21820 Oct 18 '16 at 13:53

Well, why not $V_{\beth_{\beth_{\omega_1}}}$? In fact, why not go all the way until the first fixed point? Or maybe the first fixed point which is a limit of fixed points? Or the first one which has a club of fixed points below it? Why not any of them? :P — Asaf Karagila Oct 18 '16 at 13:58

I think the more important question for people who are interested in a foundation of mathematics that models real world applicable modern mathematics and want to convince infinite set theorist is the following:

Ask them why we need a hierarchy of unreachable actual infinities in sequence and whether there are theorems that explicitly demand it with no alternatives

(The above is basically a very broad way of saying: Justify why we need uncountable well orderings and want these to be linearly ordered)

8

A: Bijection between closed uncountable subset of $\Bbb R$ and $\Bbb R$.

This is a nice result due to Cantor that is good to understand well. Let me expand André Nicolas's answer. We need a modicum of set theory, specifically, a working understanding of $\omega_1$, the first uncountable ordinal. 1. First, we need the Cantor-Baire stationary principle: Suppose tha...

hmm...

but then, the $n$-computable reals are dense but countable, so topology should not be broken when we took $\omega_1$ away from it

hmm...

Other things to think about: Non separable hilbert spaces. These don't have a countable hilbert basis. They arise in many areas in quantum mechanics and quantum field theory

so we need to investigate how will these hilbert space and their analysis changes when we replace the reals with the n-computable reals. If something too significant is lost and cannot be satisfactorily replaced, then it might justify we need some notion of uncountably. Will investigate this later

11

Q: Is there any physical system with a non-separable Hilbert space?

It seems that all the usual physical systems have a separable Hilbert space. Is there any example with a non-separable Hilbert space? BTW, I am actually always baffled by the fact that a continuous model like the 1d harmonic oscillator defined on $\mathbb{R}$ has a separable Hilbert space. It i...

quantum-mechanics mathematical-physics hilbert-space

Hmm... this is more complicated than I thought. It seems while QM and QFT will do fine by construction, some quantum gravity stuff and symmetry breaking phenomenon will run into problems. But I guess it is still too early and rare to consider them, I think...

user21820

Nov 29, 2017 17:06

Did you look at the linked post about the Lebesgue integral? Terence Tao's answer essentially shows that you need hardly anything to get the results that have real-world consequences.

That said, having the standard reals has nothing to do with ω[1].

Secret

Not yet, I just finished reading your reaplcement post

user21820

Just like V[ω[1]] has the set of countable well-orderings of N but does not have ω[1].

Secret

Actually I am thinking about something broader, not $\omega_1$: Throwing away powersets, since powersets are known to make things impredicative. This is why I talk about computable reals

user21820

When you throw away the classical power set, you still can retain the notion of function types.

This would be agnostic regarding whether func(N,bool) is uncountable in the sense of size.

So it would be compatible with both the standard reals as well as the computable reals.

Secret

Ok I see, so we basically make the (un)countability of reals ungrounded, but we can still retain most of its properties

~~I always thought things like non separability will be gone if we cannot ground the notion of uncountability~~ O wait, my brain overshoot my typing, let me reprocess...

user21820

Nov 29, 2017 17:11

You still have uncountability in the sense of complexity. No surjection from N onto func(N,bool). But without the classical power-set notion you cannot prove or disprove that func(N,bool) injects into N.

Secret

I see. Btw an unrelated question. We knew that in cardinality there's the notion of potential vs actual infinity. What are the computational theory analogues? Is it halting problems and turing jumps?

user21820

@Secret That's the usual analogous notions for potential and actual infinity, but that's nothing to do with cardinality...

Cardinality is about the existence of bijections, and in ZFC it is also about the existence of a bijection with a well-ordering.

I'll have to think about the uncountable closed subset of R. Not now though; have to go off soon.

Secret

I mean, in terms of size, potential infinity is about "increasing something forever" and actual infinity is "something that is larger than everything else"

in terms of complexity and computability, what will be the corresponding concepts?

user21820

@Secret "forever" has nothing to do with size.

Size and sequences are totally different, as I said earlier with V[ω[1]] having uncountable sets but only countable sequences.

Finite sizes and indices coincide. From ω onwards they clearly diverge.

ω+1 is longer than ω but both are the same size.

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