TonesInDeepFreeze

In set theory, equivalence does not imply equality. Here's the most trivial example: {{0 1}} is a partition of {0 1}. And that partition induces the e...

Then he's right. It takes only a moment to see that the salient feature of the argument is that it shifts from one sense of "if then" in one place to ...

Keep digging your sinkhole deeper.

I didn't ask for a definition of 'planar graph'. You didn't read what I said about this a few posts ago. You are a sinkhole.

Apparently, you don't recall the post in which I said that I'm willing to indulge you only up to the point that you go past the process of definitions...

You need to define "1D analogue of the established term "planar diagram"" in terms that don't presuppose any mathematics that you have not already def...

That relies on conflating two different senses of "if then": an everyday sense and the material conditional. I'll use '-->' for the everyday sense and...

That's BS. BS includes nonsense, doubletalk and falsity. And handwaving is not necessarily just lack of rigor to be supplied later. And you presume th...

You're conflating non-equivalent theorems. Theorem 1. The set of natural numbers is well ordered by the standard less than relation on the set of natu...

You seem not to understand how the mathematical method of handwaving works. It's not ZF or PA or one of those; it's the theory BS. You need to familia...

Thank you. You saved me a lot of time and effort. Because my prediction that you would resort to half-baked handwaving is confirmed, so I am done with...

Yes, the incompleteness theorem, when about PA in particular, is a meta-theorem about PA. If PA is consistent, then: PA |/- G PA |/- ~G PA |/- Con(PA)...

Of course, that is just sentential logic from: If PA is consistent then PA is incomplete. I'm not sure about that; I'd have to think about it. Correct...

In a given countable language, there are only countably many sentences. But there are uncountably many languages and systems, so it's trivial that the...

Here's what you wrote:

By millennia. But those concepts are not formal. You are so confused. T |= S stands for "T entails S" not "S is true in T". We know that 'true' and 'p...

You are very confused. Incompleteness is not a theorem of PA, unless PA is inconsistent. Incompleteness is ordinarily proved informally or formally in...

There are not only finitely many of them, and there are not uncountably many of them (there are only countably many sentences in the language), so the...

You keep using 'the soundness theorem' in a way that invites confusion. For the third time, the soundness theorem is: "If a sentence P is provable fro...

There are denumerably many of each.

Here's a breakdown: 'true' and 'false' here mean, respectively, 'true in the standard model for the language of PA' and 'false in the standard model f...

Where did Godel say that?

It wouldn't be used for arithmetic. But it would still have models if it is consistent. Irrelevant to what? Irrelevant to whom? It's relevant to whome...

The soundness theorem is "If a sentence P is provable from a set of sentences G, then all models of G are models of P". We don't prove that from PA. W...

There are two ways: (1) Prove a theorem from axioms. Then the theorem is true in any model in which the axioms are true. (2) Prove that the sentence i...

I accept your disclaimer. But I point out still that your comment gratuitously shifted from you to me. I asked for definitions and your retort was to ...

Like I said, I didn't get personal with you until you did with me.

Amazing that you got that exactly backwards.

You must mean that there's no point in you continuing. Others can choose for themselves.

In set theory, 'isomorphism' is not 'two things are the same that are not the same'. Rather, two things are similar; they have structures that are sim...

Just to be clear, the example doesn't assert that Venus is a star.

I put that in the category of "Don't micturate on me and tell me it's precipitation." It was a stupid comment: There's no reason to think I don't know...

I have no comment on Wittgenstein.

My posts were about correcting misstatements about the theorem and about the lack of clarity in your errant arguments about reactions to the theorem. ...

You don't know what you're talking about: (1) Of course I know the halting problem. You make no point with the false insinuation that I don't. (2) I k...

I wouldn't say that. (1) the set has only three members, not four, (2) the legs of the horse is not that set; rather the cardinality of the set of leg...

It seems that most often I'm not patient. In these situations, patience is a fault.

Two different senses of 'complete': (1) a theory T is complete iff for every sentence P in the language of T, either P is a theorem of T or ~P is a th...

It's not clear to me whether you're suggesting that my remarks were not pertinent. But in case you are: The conversation has had many subjects. You me...

What are the definitions of 'complete for a complexity class size' and 'consistent for a complexity class size' such that a logic can be complete and ...

How so? How would such a thing show that it is not the case that every consistent, formal theory sufficient for arithmetic is incomplete?

What's another? If it's just Hilbert's program, then why not just say that from the start? But then your reasoning about that limitation in connection...

Depends on what you mean by "everything". We were talking about incompleteness, in which context 'complete' and 'consistent' have certain definitions....

So one of the conclusions you are referring to is "incompleteness puts a hard limit on understanding the world"? Are there any writers who you think a...

What are complete and consistent, or incomplete or inconsistent are theories, not whatever "decidability for P v NP" is supposed to mean.

It is taken that incompleteness quashes Hilbert's program. I think it might depend on one's definition of 'logicism' whether incompleteness also quash...

What do you mean by 'decidable'? 'decidable' in context of incompleteness and computability theory has a definition. Is yours the same?

I don't know what that is supposed to mean. But incompleteness is not avoided by greater and greater proof capability. No matter how capable, if the s...

I don't know what that is supposed to mean, but, to be clear, the incompleteness theorem applies also to theories in higher order logic. Indeed, Godel...

Maybe you mean Rosser. Rosser improved Godel's theorem, but that has nothing to do with what I said in my post. That is very wrong and backwards. No c...