What are the mathematical definitions of

separation
dimensional separation
dimensional point
non-dimensional point

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 outputs) can't do it.

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From my non-exhaustive readings on Turing machines, I don't know what an 'error state' is for a Turing machine. Basically, a Turing machine takes an input and either halts with a state and whatever is on its tape when halted or it does not halt.

But there are different formulations of Turing machines, so I would welcome knowing of one that defines an error state.

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 is supposed to be the inherent order of the set whose members are all and only the bandmates in the Beatles. Without an answer to the question, the claim that every set has exactly one inherent ordering is unsustained. I've asked the question many times, but it has not once been addressed.

Specific article or text? What specific theorem of mathematics is used to derive physics that's mixed up?

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 been banned.

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 defined terminology in the literature.

It was claimed that certain ideas in physics are mixed up because of importation of certain mathematics. What are some specific examples of published work in that regard?

Mathematicians don't claim that the mathematical sense of 'countable' corresponds to the everyday sense of counting a finite number of objects. The use in mathematics is a certain technical sense:

S is countable if and only if either S is finite or there is 1-1 correspondence between S and the set of natural numbers.

And in that mathematical context it is not the case that 'infinite' and 'uncountable' mean the same.

Technical fields of study often have special definitions. Quibbling about that is pointless.

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 real numbers and L is the standard ordering of the set of real numbers. If we speak of a "line" then perhaps, to be most exact, we would consider the set {<x 0> } | x is a real number}. (In other words, the "horizontal axis".)

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 defined in terms of 'gaps', nor 'executions', nor algorithms.

'Dedekind cut' is mathematically defined and we see mathematical proofs that the set of Dedekind cuts is uncountable.

And it is serious misunderstanding to argue that there can't be more Dedekind cuts than there are rationals. A Dedekind cut is a certain kind of pair of subsets of the set of rational numbers. The set of rational numbers is countable, but the set of subsets of the set of rational numbers is uncountable. A person who argues from the fact that there are only countably many rationals to the claim that therefore there can be only countably many Dedekind cuts is a person who does not know what a Dedekind cut is.

Another poster mentioned "medical propaganda" regarding bird flu. What are some specific examples that are claimed to be medical propaganda regarding avian flu? Specific medical institutions posting propaganda or disinformation about avian flu? Specific news outlets (such as AP, Reuters, NBC, CBS, ABC, CNN, NYT, WP)?

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It was claimed that Amish communities weren't hit hard by Covid. I take that to mean that among people who lived in Amish communities there were starkly lower rates of infection, painful sickeness, hospitalization, death and long Covid. What is the source for that claim? (Or was the claim merely a fanciful joke premise so that both the setup and punchline are not presumed factual?) The punchline was that Amish communities didn't suffer so much because they avoided media coverage of the pandemic. Even though the context is that of a joke, is it nevertheless being suggested that knowing less about the pandemic has a causal relation in reducing infections, painful sickness, hospitalization, death and long Covid?

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 disjoint intervals.

Ostensively:

(0 1/2)
[1/2 3/4)
[3/4 7/8)
...

Formally:

Let f be the function whose domain is the set of natural numbers such that:

f(0) = (0 1/2)
for n>0, f(n) = [(2^n - 1 )/2^n (2^n+1 - 1)/2^n+1)

The range of f is an infinite partition of (0 1). That is: the range of f is infinite; every member of the range of f is an interval; the range of f is pairwise disjoint, and the union of the range of f is (0 1).

Back to the poster who claims to offer an alternative to classical mathematics: The word 'isolate' keeps coming up. What is a rigorous mathematical definition of 'isolate'?

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, there is no first order theory whose models are all and only the complete ordered fields. Yes, in set theory we define 'complete ordered field' (some people call the clauses in the definition 'axioms') and show that all complete ordered fields are isomorphic, but that is different from a first order theory whose models are all and only the complete ordered fields.

There are first order categorial theories of such things as real closed ordered fields, but they do not include the completeness property that is crucial for an adequate account of the real numbers.

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 paradox occurs?

In classical mathematics, computable reals are not the programs for computing their digits nor the equivalence class of such programs.

The number and the program are different things. The number and the equivalence class of programs are different things.

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 domain is not infinite, then n cannot be arbitrarily large.

One is welcome to work it out in some other way. But then the natural question is: What are your primitives, formation rules, axioms and inference rules? Mathematicians give us the courtesy of stating those rigorously so that it is utterly objective, by machine-verification, whether a purported formal proof is indeed a formal proof. Or, one could say that one doesn't do things formally. That's fine, but then a comparison with mathematics is not apt since mathematics rises to a challenge that informal quasi-mathematical ruminations do not.

.3... is not a program; it is a number.

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'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 sum or infinite sum.

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Mathematicians have not been sloppy in distinguishing rationals from reals. Mathematics uses a rigorous method definitions. The definitions of 'is a rational number' and 'is a real number' are completely rigorous. Suggestion to anyone not familiar with that fact: Read any one of a number of books in which the constructions of the rationals and of the reals are shown step by rigorous step.

"compute to infinitely fine precision" requires a mathematical definition. A mathematical definition would be of the form:

A program P computes a real number x to infinitely fine precision ifff F [where F is a formula whose only free variable is x and that uses only previously defined terminology]

Meanwhile, along the lines mentioned:

(1) For any computable real x, there is a program that lists the digits of x and does not halt so that any step in execution, only finitely many digits have been listed.

(2) For any computable real x, there is a program whose domain is the set of natural numbers, and for any n, the program outputs the nth digit of x and halts.

(3) The set of digits in the decimal expansion of 1/3 is the singleton {3}, thus a computable set.

That's mathematics. For "compute to infinitely fine precision" to be mathematics, it requires a mathematical definition.
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Mathematicians do not "obfuscate" programs with executions of programs. If one claims that mathematicians do that, then what are specific written examples? If there are no specific written examples, then it's a strawman.

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Classical mathematics does not call program 'numbers'. (However, programs can be assigned Godel-numbers.)

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Mathematics uses a specific method for definitions. Alternative definitions would themselves need to adhere to the methodology of definitions which includes the criteria of eliminability and non-creativity, which insure against vagueness, circularity, and infinite regress, most pointedly that the definiens use only primitives or previously defined terminology.

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This is a non sequitur: We do not physically experience infinitudes therefore the existence of infinite sets must be derived and not axiomatic.

If one proposes a mathematics without infinite sets, then that is fine, but the ordinary mathematics for the sciences uses infinite sets, which are not derivable from the rest of the set theoretic axioms, thus requiring an axiom.

For that matter, we don't physically experience breadthlessness, so breadthlessness is itself an idealization, just as infinitude is an idealization.

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The notion of length is quite coherent. In context of the reals, length is a property of segments not of points. And we have a perfectly rigorous explanation of how it works: Let x and y be two different points, the length of the segment with x and y as endpoints is |x-y|. This is a notion that is clear not just to mathematicians but to school children.

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Who said anything equivalent with "the continuum constructed in this fashion is paradoxically beautiful and only to be seriously discussed by the experts"? Specific quotes are called for. Otherwise the claim is a flagrant strawman.

"Really?" No, really, who said anything equivalent with "the continuum constructed in this fashion is paradoxically beautiful and only to be seriously discussed by the experts"?

Whatever the shortcomings of the poster, at least he demonstrates skill in the use of Ctrl-C and Ctrl-V.

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 preserving.

And these points that the poster lacks focus to understand:

Contrary to the poster's confusion, the principle of explosion accords with truth preservation.

Contrary to the poster's ignorance, classical logic handles the notion of truth in terms of correspondence with states of affairs.

Contrary to the poster's addlement, truth and entailment are different notions. A set of premises entails a conclusion if and only if there are no states of affairs in which the premises are all true and the conclusion is false. Whether or not the conclusion is or is not false is a different question.

Contrary to the poster's ineducation and lack of intellectual curiosity, there is an approach to logic called 'relevance logic' in which the conditional is handled with regard to the contentual features of the antecedent and consequent, as that study is, unlike the poster's pronouncements, informed and rigorous.

Contrary to the poster's intellectual recalcitrance, merely to refute the poster's falsehoods about classical logic, one is not required to indulge the poster in his confused notions about some claimed new foundation he thinks he has created.

That the poster is monomaniacal about some pet idea of his own doesn't entail that anyone else who points out his copious errors about classical logic lacks focus for not nodding in acceptance of his spammy gibberish.

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 refuses to understand the matter. To explain that classical logic is truth preserving, one doesn't have to affirm the poster's grandiose vision that from the mountain top he is bringing to the world the stone tablets of a new foundation.

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 G nor ~G. That is the very statement of incompleteness.

The poster not only fails to desist from (let alone retract) his previous lie that I have not addressed the subject substantively, but he reposts it in all capital letters.

The poster responds yet again to refutations by merely reposting his pet statements. Either the poster suffers from repetition compulsion or the poster is a bot.

And True(PA, G) has no apparent meaning.

A sentence is true or not per a model, not per a theory. Though, of course, a sentence may be true in all models of a theory, which reduces to the sentence being a theorem of the theory.

Ah, another fallacy from the poster. The fact that personal comments are added to substantive comments does not entail that the substantive comments have not been amply given.

I've given exact detailed technical explanations of how the poster has been wrong in so many ways. Discussing also that the poster is a crank does not erase the technical explanations.

The poster is flat out lying that I have not shown errors in his reasoning. My refutations have been copious.

And notice that the poster yet again evades dealing with the very quote of mine that he posts. To wit:

The poster's reasoning includes giving evidence that Wikipedia's article on the Peano axioms does not mention Godel numbering and diagonalization. But (1) The axioms don't mention a lot of things, not even such things as the fundamental theorem of arithmetic. The axioms don't mention Godel numbering and diagonalization but that doesn't entail that they are not basically applications of arithmetic. Indeed, finitistic arithmetic. (2) The Wikipedia article actually DOES discuss the incompleteness theorem regarding PA.

And that is just the latest in a long chain of evasions and misrepresentations by the poster, not even counting all the other redundant threads he's propagated on these subjects.

And his latest message misses the point also:

The question is not ascertaining what is true. Mathematical logic has a rigorous definition of 'true' in which sentences are evaluated first with their atomic components such that an atomic sentence is true if and only if it corresponds to the given state of affairs.

Rather, the question has been about entailment. And the principle of explosion doesn't say that the conclusion is true, only that a contradiction entails any statement. That is, since there are no states of affairs in which a contradiction is true, there are no states of affairs in which a contradiction and any other sentence are together true. Regardless of how we reckon the truth of atomic statements, there are no circumstances in which a contradiction is true thus no circumstances in which both a contradiction and any conclusion are both true.

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Getting back to the key point that started this part of the discussion:

The poster claims that classical logic is not truth preserving. But it is, as I have explained (and it is proven). And the fact that the poster prefers his own vague, undefined and confused outlook on logic does not entail that classical logic is not truth preserving.

Truth preservation is: If the premises are true then the conclusion is true. And that is PROVABLY upheld by classical logic.

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 any point at all here!

A specification of PA itself does not include mention of all kinds of results in arithmetic, from the fact that there is no greatest prime to the fundamental theorem of arithmetic to the most advanced and complicated theorems about the natural numbers ... and to Godel-numbering and diagonalization. So what? It doesn't entail that those developments are not provided for, or pertain to PA.

Here's a pretty good analogy: The rules of chess make no mention of the incredibly complicated strategies of chess masters, but that does not entail that those strategies are not permitted by the rules nor that analysis of those strategies does not pertain to chess.

And as the poster touts the Wikipedia article, he conveniently omits including that the article itself DOES mention the incompleteness theorem!

What a seriously risible argument the poster makes! Really, the poster is as hopelessly ignorant, confused and irrational as they come. I've seen some that are more dishonest, but the poster ranks fairly high in dishonesty too, as just witnessed that he touts a Wikipedia article that actually shows the OPPOSITE of his own claim!

The poster says that understanding that a sentence can be both untrue but not false requires knowing about his own undefined "truth-maker maximalism".

More pertinent is that understanding this subject requires knowing, as the poster does not, at least the basics of symbolic logic with the basics of mathematical logic to follow.

But that is the way of the crank: The requirement that everyone else accept the crank's undefined terminology, impressionistic musings, and illogical arguments while the crank himself has no responsibility to learn even page one of an introductory textbook.

The poster said that G is untrue. Now he says he did not say it is false.

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Godel numbering and diagonalization use only arithmetic. Godel numbering and diagonalization use arithmetical operations in very complicated and surprising ways, but nothing beyond arithmetic is used. When one actually reads the proofs, one sees that each step is utterly unassailable mathematics, well within arithmetic even if a complicated and ingenious use of arithmetic. Indeed, the proof can be done in finitistic intuitionistically acceptable arithmetic. If there is a context that is epistemologically safer than that, then I'd like to know what it is.

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-sentence is true.

It is completely a confused notion that G is false.

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If the poser were sincere about discovery in mathematics, he'd use his own terminology and define it, rather than piling on confusion by conflation with standard terminology. Let alone that he'd sit himself down to learn the basics of the subject starting with a textbook in introductory symbolic logic rather than misconstruing and misrepresenting stray bits of poorly written and intellectually disorganized Wikipedia quotes.

And, just as I discussed, there we have the poster saying that he's a source of brand new ideas while all his interlocuters are wrong: the fantasy that he's a brilliant maverick math hero. And The Philosophy Forum is just one of the Internet forums that he uses to act out his fantasy.

But I applaud The Philosophy Forum for its toleration, allowing even the most incorrigible cranks to start thread after redundant thread, spewing disinformation like a crudely written bot.

Thank you, fdrake, for those useful words.

Yes, 'epistemological antinomy' is not mentioned as a formal mathematical rubric in Godel's famous paper. The poster abysmally fails to read - fails even to recognize that it has been pointed out to him - that Godel confines the significance to that of analogy not of formal application.

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 basic aspects of this subject thus making sure that he is not overwhelmed by facts and logic.

Again, the answer begins in the post to which the poster was replying and then elaborated on in my next post. In hopes that it is not ignored yet again:

'shows' in this context does not mean that there is contentual relationship but merely that there are no circumstances in which all the premises are true and the conclusion is false. For an alternative, one would study relevance logics.

The poster's earlier claim that started this part of the discussion was that classical logic is not truth preserving. But it is truth preserving since there are no permitted inferences from true premises to false conclusions. That is the case no matter that classical logic is not a relevance logic.

The poster asks a question anew. He should read the post to which he is replying.

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 context does not mean that there is contentual relationship but merely that there are no circumstances in which all the premises are true and the conclusion is false. For an alternative, one would study relevance logics.

The poster's earlier claim that started this part of the discussion was that classical logic is not truth preserving. But it is truth preserving since there are no permitted inferences from true premises to false conclusions. That is the case no matter that classical logic is not a relevance logic.

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Still interested to know the basis for the claim that most philosophers reject the analytic-synthetic distinction.

Still interested whether the poster will ever admit that he improperly conflated contradiction with falsehood by overlooking that, while contradiction implies falsehood, falsehood does not imply contradiction.

Still interested whether the poster now understands that a footnote is in context of the passage to which it is a footnote.

And meanwhile, the poster still has not understood that he gets both Godel and Tarski exactly backwards. Most specifically that Godel does not claim that the system proves that the there is a proof in the system of the Godel-sentence, but instead proves that the system does not prove it; and Tarski does not use the liar sentence as a premise in any proof (as "this sentence is not provable" is crucially different from "this sentenced is false") but instead proves that the liar sentence cannot be formed in the language.

How to explain cranks? They take a position that they have a system for mathematics that is correct to the exclusion of mathematicians who are terribly wrong. In order to make an impression with that position, the crank postures that he dismantles the work of the mathematicians. But in order to carry out that dismantling, the crank must get the mathematics quite wrong. Not only is the crank's supposed system vague, impressionistic, illogical (to the point of being incoherent) and uninformed, but the crank's remarks about the work of mathematicians are woefully ignorant, confused and disinformational while self-fortified against being corrected on virtually any point ranging from fundamental to incidental, no matter how utterly clear it is that the corrections are sound. And sometimes this goes on for literally decades of spammed repetitions. The only explanation I can think of is that the crank's need is not to understand mathematics or even well conceived alternative mathematics, but rather the crank has a deep need to be taken to be an exceptional, remarkable person who single-handedly has debunked the work of mathematicians. The crank seems to want to live out a kind of hero fantasy and "triumph over conformity" fantasy no matter the facts or logic about the subject, no matter that thereby he makes an abysmal fool of himself though, of course, not in his own hopelessly blinkered mind.

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 by simply reading what I had posted many posts ago. What is exhausted is his own attention span that exhausts itself in less than a minute.

As to honest dialogue, there is no point of dishonesty that the poster can point to in anything I've written, while meanwhile, only a few posts ago, the poster tried to evade a key point by dishonestly conflating contradiction with falsehood, as he continues to dishonestly evade that point.

As to the claim that I am as disagreeable as possible, there is so much confused disinformation written by the poster that there is indeed a great amount to disagree with.

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The poster cites "semantic connection". That is not a defined term. However, the semantics are clear, as I have mentioned over and over but the poster refuses to recognize:

The notion of semantic entailment is:

A set of premises entails a conclusion if and only if there are no circumstances in which all the premises are true and the conclusion is false.

If the premises include a contradiction, then there are no circumstances in which all the premises are true and the conclusion is false. Therefore, if the premises include a contradiction, then those premises entail any conclusion.

By the way, mutatis mutandis it works the other way too: If the conclusion is logically true, then that conclusion is entailed by any set of premises.

Little doubt the poster still does not understand these points. He was too exhausted to understand them decades ago when he began not learning even the basics of this subject.

On the other hand, there is an alternative approach to logic call 'relevance logic' that does formalize the notion of the content of sentences and bases a different notion of entailment in that regard. And the advantage of that work, in contradistinction with the confused, ignorant and intractable handwaving of cranks is that the work is (as far as I know) rigorous and it is grounded in clear understanding of the subject - both classical and alternative. Some people prefer such approaches as relevance logic though they are more complicated than plain classical logic. But the existence of alternative approaches does not refute that my own reports of classical logic have been correct. And the takeaway here is that study of alternative logics requires prior understanding of the most basic logic that the alternatives include in some parts, reject in some parts and extend in certain ways. We need to be correct in what we say about classical logic if we are to properly critique it or propose a supposedly remedial alternative to it.

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 is central to the matter being discussed.

Meanwhile, yes of course as I have agreed at least a dozen times by now and as in the very post he just now responded to (!):

If C is any contradiction and Q is any sentence, then:

C |- Q

That is a basic result in sentential logic, known to anyone who has studied the subject.

But the poster just keeps posting it over and over though no one disagrees with it.

He might as well say, "2+2 = 4. Ha! Take that!"

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 from the inconsistency of the premise to the falsehood of the premise.

But to address the latest post anyway:

Let C be a sentential letter and, for some sentence P, we define: C <-> (P & ~P).

Then for any sentence Q:

C |- Q

But since we don't have C, so we don't have Q.

And of course, in any model C is false, while depending on Q, in a given model, Q could be true or false.

Getting back to the poster slipping from the context of contradiction to falsehood: Yes, all contradictions are falsehoods. But not all falsehoods are contradictions. The point here is that mere falsehood is not what's involved in the principle of explosion.

When the poster tries to justify the slip by noting that all contradictions are falsehoods, he commits the illogic of getting the matter backwards: It's not a matter of all contradictions being falsehoods (which is true) but rather it's that the converse does not hold. The poster can't reason himself out of the proverbial paper bag.

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 contradiction and thus not true.

But if any of the sentences in the set are quantified, then the set might be inconsistent (yielding a contradiction) though we don't know it's inconsistent, and a given system might not provide a mechanical means to verify whether a given set of sentences is inconsistent or not.

Moreover, the principle of explosion regards entailment, no matter what we happen to know about the truth, falsehood or inconsistency of the sentences. The principle of entailment, even in ancient form, concerns the impossibility of states of affairs whatever our knowledge or lack of knowledge about those states of affairs.

Notice that the poster switched the discussion from the premise being contradictory to the premise being merely false. We had been explicitly discussing contradictory premises not just false premises.

Consider two sentences D and M.

If D is false, then M might be true or false. The falsity of D does not entail the truth of M. But the conditional D -> M is true.

If D is a contradiction, then M might be true or false. From the mere fact that D is logically impossible we do NOT infer that M is true. But the conditional D -> M is not just true, it is logically true. THAT is the principle of explosion and it does NOT imply that M is true.

If the sentence D is merely false, we cannot infer M. But if we assume the sentence D & ~D then we can infer M. But, again, we would be mistaken to have D & ~D as a premise, so if M is false, thus a mistake to assert it, the mistake of asserting M would occur only by the mistake of asserting D & ~D.

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 the formalization of the idea.

And model theory adheres to the ancient notion of entailment: A set of premises entails a conclusion if and only if there are no circumstances in which all the premises are true but the conclusion is false.