TonesInDeepFreeze

As I said, the incompleteness theorem pertains only to recursively axiomatizable theories. Montague grammar does not in and of itself make a general s...

Not a finite set of axioms, rather a countably infinite set of axioms. However, indeed, not an uncountable set of axioms as was incorrectly claimed by...

The incompleteness theorem applies to formal theories, not to any set that includes natural language expressions. Moreover, the incompleteness theorem...

I don't have anything immediate to say about people's subjective impressions of mathematics. Q is enumerable and so is the set of finite sequences of ...

I don't see it as mystic or foggy.

The one constructed in the proof. When you read the proof, you see the G that is constructed. Of course you're free to investigate whatever you like. ...

Thanks, jgill. Good to see you too. Probably only a brief stopover for me though. My lifecoach pandemic response guru (just kidding) tells me that my ...

Thank you for that. I am not an expert, but sometimes I have good (rigorous) understanding of some of the basics, though, over time I've become rusty ...

To be clear, the problem is not people who can't understand the mathematics, but those (not necessarily ones lately in this thread) who refuse (throug...

The basic idea of what the theorem says can be stated roughly in common language: If T is a consistent theory that expresses basic arithmetic, then th...

Just to be clear, some consistent systems are incomplete while others are complete.

Forget about the idea that's been presented in this thread, in connection with incompleteness, of numbers referring to themselves.

If I'm not mistaken, the only authoritative translation is the one in 'From Frege To Godel'.

It's not a question of "natural". It is utterly rigorous though. We define a certain function from the set of symbols into the set of natural numbers....

Sure there is. Just read a textbook on the subject. But if you're not interested in doing that, then indeed there's little hope that you'll understand...

The completeness theorem is not a mere conjecture. It is a theorem of mathematics. For any given consistent theory in a countable language, the proof ...

Yes, the one that we show in the proof of incompleteness. By the way, it is not undecidable in every theory. It is undecidable in particular theories....

No. As an illustration, it should be "The statement on the list at location n is not provable" and the location on the list for that statement is n. G...

You need to read a textbook. Then you could ask about what you don't understand. A few chats with a mathematician might give you an overview, but to a...

G is not a number. G is a formula. Then G is also coded by a number, which is its Godel number. The formula G and the number that codes G are two diff...

And the incompleteness proof doesn't say "this sentence". It's more complicated than that, and thoroughly rigorous.

I'll pick some highlights in this thread. Either G is provable in T or G is not provable in T. No.

In context of proofs of incompleteness, we don't have numbers referring to themselves. Rather, numbers are used to refer to symbols, to strings of sym...

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Good post. Interesting. I'd need to read that essay, but meanwhile, what does he mean by "fundamental logical notion"? And why does he say there is no...

I think Russell is saying that you don't put modal operators in front of terms, only in front of formulas. So you could have: NEx(Kx & Ay(Ky -> y=x) &...

Copleston seemed not to be familiar with the thinking behind Russell's point that he eschews 'necessary' and 'contingent' as adjectives to describe en...

In this thread? Which posters do you claim are limited in their thinking to only physical reality? And what have they posted that allows you that clai...

Most importantly, it is the way to confirm (to yourself or to anybody) that you have actual mathematics free of any hand-waving.

Not a lot of money. A few good books. Right. No Kate and Edith too. Yes, and down that road we arrive at a concept free of the paradoxes of naive set ...

Interesting dialogue at that link.

Famously, it's a given that sophistry, intellectual dishonesty, downright factual dishonesty and confusions born of ignorance are rampant on the Inter...

I didn't say that I'll only consider formalizations. I have been interested in the earlier proposals though not formalized. Rather, I said that I'm no...

Laughter is not the involuntary response provoked here.

In that post, Gnomon makes a number of statements about mathematics qua mathematics. Attempting defintitions (thery're botched) and making claims. Abo...

You say you'll do it again. And you did just do it again. The bolding you put in was not mine. And you did it just to prove that you can - a childish,...

Seems that I didn't include my reply to that. It is not clamed that mathematics solves those philosophical problems. It is only claimed that mathemati...

Of course, the Greeks were mathematical pioneers, but they had not formed a mathematically satisfactory account of the notion of infinitude. That does...

This also should not have been SKIPPED over:

You did it again: You bolded in my quote without indicating that the bolding was not original. I have asked you at least three times not to do that, b...

I just wanted to know whether you understand. I like nodes better than paths for this. In set theory, every denumerable sequence of nodes converges to...

PERSONAL MOTIVATION What attracted me to the S-B tree in this thread is that we can take reals to be sequences of nodes. Unlike with equivalence class...

The bolding is not mine. You've quoted me a few times as you've added bolding. I already asked you to note that the bold or emphasis is added. Are you...

Let C = 1. Pi = C/D. So 1/Pi = D Toward a contradiction, suppose D is rational. So there are integers n and m (m not equal to 0) such that D = n/m. So...

These stand: /

That is nothing less than bizarre for you to say. You wrote: And I said that you wrote: "Infinite Regress" and "First Cause" are philosophical concept...

You are still confused. You still SKIP the MAIN points I post. You SKIP over the explanations about how you're mathematically wrong (not wrong to esch...

So what? Lots of things use similar language, but say RADICALLY different things.

What? I listed the CRUCIAL, ESSENTIAL ways in which they are different. Rather than recognize that, you cop out with "it's a matter of perspective wha...

You haven't even figured out the first "dimension". But carry on, though I will very likely not be subscribing.