Not only cannot a physical computer produce an infinite string, but a Turing machine (not an extension of the notion of a Turing machine to infinite o...
By the way, related to the claim that mathematics is fundamentally errant by its notion of sets without inherent order, I'm still interested in what i...
I very much appreciate the positive remarks sent my way. But, for what it is worth, which is not much, my position is that the poster should not have ...
CORRECTION: I was incorrect when I said that "truth-maker maximalism" is the poster's own undefined terminology. But later I found out that it is a de...
It was claimed that certain ideas in physics are mixed up because of importation of certain mathematics. What are some specific examples of published ...
Mathematicians don't claim that the mathematical sense of 'countable' corresponds to the everyday sense of counting a finite number of objects. The us...
Another poster said that the real line is the set of real numbers. Just to be exact: The continuum is the ordered pair <R L> where R is the set of rea...
Back to the poster who doesn't understand the basics of this subject. 'gap' is mentioned but not defined nor is 'execute a cut'. 'Dedekind cut' is not...
Another poster mentioned "medical propaganda" regarding bird flu. What are some specific examples that are claimed to be medical propaganda regarding ...
It was claimed that the interval (0 1) is not an infinite union of disjoint intervals. It is false that the interval (0 1) is not an infinite union of...
Back to the poster who claims to offer an alternative to classical mathematics: The word 'isolate' keeps coming up. What is a rigorous mathematical de...
Another poster mentioned a categorical theory for the reals. Just to be clear: There is no first order theory of complete ordered fields. That is, the...
It was claimed that Russell's paradox is "still there". In what specific post-Fregean systems is it claimed that the contradiction of Russell's parado...
In classical mathematics, computable reals are not the programs for computing their digits nor the equivalence class of such programs. The number and ...
It has been proposed in this thread that a sequence converges as n gets arbitrarily large. A sequence is a function. A function has a domain. If the d...
.3... is not a program; it is a number. / '1.0' not= '1.0...' 1.0 = 1.0... '1.0' and '1.0...' name the same number, whether is is named as a finite su...
"compute to infinitely fine precision" requires a mathematical definition. A mathematical definition would be of the form: A program P computes a real...
I am precisely focused on the central point from which this discussion is pursuant: Contrary to the poster's false claim, classical logic is truth pre...
Whatever else the poster thinks he is doing, he claimed that classical logic is not truth preserving. I explained why that is false. The poster still ...
The poster writes two sentences that are lies. The poster lies that I believe that PA proves G or it proves ~G. It is the opposite. PA proves neither ...
The poster responds yet again to refutations by merely reposting his pet statements. Either the poster suffers from repetition compulsion or the poste...
Ah, another fallacy from the poster. The fact that personal comments are added to substantive comments does not entail that the substantive comments h...
And the poster links to a Wikipedia specification of PA and points out that it does not include Godel-numbering or diagonalization. As if that makes a...
The poster says that understanding that a sentence can be both untrue but not false requires knowing about his own undefined "truth-maker maximalism"....
The poster said that G is untrue. Now he says he did not say it is false. / Godel numbering and diagonalization use only arithmetic. Godel numbering a...
The Godel-sentence G is proven true in a meta-theory that is ordinary arithmetic. It is not at all controversial that in plain arithmetic the Godel-se...
Thank you, fdrake, for those useful words. Yes, 'epistemological antinomy' is not mentioned as a formal mathematical rubric in Godel's famous paper. T...
I am not overwhelmed by the details here; I am addressing them and contributing them. The poster though ignores not just the details but the most basi...
The poster asks a question anew. He should read the post to which he is replying. But I'll say it again in yet different terms: 'shows' in this contex...
The poster says he's exhausted by the time it took to establish that a contradiction implies any sentence. He could have saved himself that exhaustion...
The poster still refuses to recognize that he tried to evade a key point by conflating contradiction and falsehood. That is not a mere detail, but it ...
The poster got it backwards again! It's not a matter of the conclusion being false but rather that the poster previously tried to slip the discussion ...
Getting back to the poster slipping from the context of contradiction to falsehood: Yes, all contradictions are falsehoods. But not all falsehoods are...
If D is a contradiction of the explicit form P & ~P (or any purely sentential form or even monadic form), then we can mechanically verify that D is a ...
Notice that the poster switched the discussion from the premise being contradictory to the premise being merely false. We had been explicitly discussi...
The undefinability theorem is from before the method of models was given a fully formal definition. But the basic idea of a model was used long before...
This would be amazing if we didn't know the poster's history: I asked for the basis for the claim. The poster replies by merely repeating the claim. /...
I merely defined two different functions with two different domains. x is the last value in the sense that it is the value of the last member of the d...
It was claimed in this thread that most philosophers believe it is not the case that there are sentences that are true on the basis of their meaning. ...
By the way, that Wikipedia article is another example of misinformation in Wikipedia. We don't need to bring in the question of knowledge in this cont...
The ordinary distinction between truth and validity is: A sentence is true or not per a given model. A sentence is valid if and only if it is true in ...
The principle of entailment goes very far back in the history of logic. It is in model theory that the principle is given mathematical exactness. The ...
Rather than merely bandying Richard Montague, the poster would do well to start at the beginning with symbolic logic as presented in his textbook: Log...
* 'entailment' and 'consequence' are usually taken as specifying the same relation. That is the relation between a sets of sentences G and a sentences...
The meanings of sentences are given by the method of models. The truth or falsehood of sentences is determined by rules operating on the truth and fal...
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