Whether one agrees with Michael or not, at least he has been making a good faith argument and refining it along the way. And fishfry pertinently prese...
The butting of heads over Benacerraf can be reduced at least somewhat if we look closely at the premises. Two options: (1) We do not make explicit the...
The use of 'can' there is merely colloquial. We may state it plainly: Any set of sentences is a set of axioms. More formally: For all S, if S is a set...
It doesn't matter whether we're proving that there is no list of all the reals, or no list of all the reals between 0 and 1, or (as in Cantor's proof)...
I have an idea that may help pedagogically. In discussions about languages, models and theories, the prepositions 'for' and 'of' might get overlooked ...
I find that quote quite understandable, quite clear and not confusing. In a meta-theory we define 'is a model' and we talk about models for languages ...
The Enderton reference was to the identity axioms. See page 112 in the logic book. And also, on page 83, he specifies satisfaction regarding '=' so th...
Classical mathematics is regarded as being formalized by ZFC. ZFC starts with a base of first order logic with identity. Whether called 'first order l...
I've never seen one. Every one of the hundreds and hundreds of cranks I've seen lacks the self-awareness to understand the ways in which they are a cr...
ZFC is a set of sentences. Any consistent set of sentences has models, and the universe of a model is a non-empty set. There is no incapability of dis...
Set theory says what is the requirement for being a model for set theory is. But set theory (if it is consistent) does not claim that there is model t...
No, theories and interpretations are different things. But, yes, I did give reasoning by which there are uncountably many theories and uncountably man...
Of course we don't need any axioms to do a whole bunch of arithmetic. But just doing a bunch of arithmetical computations is not, in the context of ma...
As I only perused, there's a writeup there about carrying out arithmetic with logic gates. But to say that arithmetic can be reduced to logic requires...
They ordinary way of writing it up is quite understandable, in context of Godel's original paper and in context of just about any article or book on i...
Of course, we understand that computations are finite. But the specific mathematical statement you made earlier was incorrect. You'd do yourself a fav...
My knowledge of mathematics, logic and philosophy is quite meager. But I do have a good grasp of certain basics and an intent not to misstate them (at...
I don't know that it was a habit, but of course he made his agenda prominent. But that is a far cry from insisting that all his colleagues work on it....
I would not write it that way. But, yes, my point is that the existential quantifier is in the meta-language and not in the scope of the turnstile. Wh...
As I understand, we agree. Godel gave an outline that leaves out needed details. But you said he was "slippery". I don't see what is slippery about it...
A proof is a sequence of sentences such that every sentence is either an axiom or follows by a rule of inference from previous sentences in the sequen...
More childishness from you. From the fact that I'm not interested in going over all your stuff all over again, it is not entailed that I rule out bein...
I didn't respond to your notion of an algorithm, since it doesn't vitiate that (0 1) is an infinite disjoint union of intervals. Again, an example of ...
It doesn't matter whether I like you. For that matter, I can't have any fair opinion of you as a person aside from this extremely narrow context of po...
As I said much earlier in this thread, it is the first order theory axiomatized by: Axiom: Ax x = x (law of identity) Axiom schema (I'm leaving out so...
I did not say that. I said that classical mathematics has the law of identity as an axiom and that classical mathematics abides by the law of identity...
They are included in classical mathematics. They may be developed in set theory, without using the axiom of infinity, but they are still included in s...
What makes them cranks is not that they don't accept that there are infinite sets nor that they find the notion of infinite sets nonsense or fatally p...
In classical set theory, we define 'is infinite'. I don't know whether there is a definition of 'is potentially infinite' in any theory. But, it seems...
He proposed the project. But he insisted that all of them undertake it? Moreover, is there even one colleague to whom Hilbert insisted the colleague u...
Do you mean the diagonalization lemma applied to the negation of the provability predicate? The diagonalization lemma doesn't have an existential quan...
Some of the elements of B are infinite. Those members of B that are infinite don't have a finite conjunction, but all natural language expressions are...
The second completeness theorem is: If S is a formal, consistent theory that adequate for certain arithmetic, then S does not prove the consistency of...
I think the theorem you have in mind is that there is no algorithm that decides whether a program and input halt. The proof uses diagonalization. But,...
No, diagonalization does not require indirect proof. No, diagonalization does not require indirect proof. Proof of incompleteness is usually construct...
Not in mathematical logic. A sentence is provable from a set of axioms and set of inference rules if and only if there is a proof sequence (or tree, t...
Given a particular countable language and meta-theory with a countable alphabet: This is correct: Given a countable set of symbols, there are exactly ...
Given a theory adequate for a certain amount of arithmetic, for example, PA, it's redundant to say that the theory proves all its theorems. But if the...
It is a false meme that the Cantor proofs mentioned here are by contradiction or indirect. Moreover, the proofs are constructive. (1) The theorem know...
The crank needs to follow the conversation but first he needs to learn some basic mathematics. The matter under consideration is whether time can be d...
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