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GrandMinnow last won the day on February 26

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  1. Modern logic concerns itself with both syntax (which is purely formal) and semantics (meanings of the expressions). / 'valid' may mean different things in context (varying among different authors in modern logic). But usually, 'valid' does not refer merely to whether the expression is obtained according to the syntax rules (the expression is well formed), but rather to whether the expression is a formula that is true in all interpretations of the language. So the notion you are referring to is more usually known as 'well-formedness', while the notion of ''validity' refers to semantics.
  2. Ordinarily it is not said that a conclusion preserves truth, but rather that an inference rule does or does not preserve truth. An inference rule is truth preserving if and only if whenever the rule is applied to true premises it yields a true conclusion. Those inference rules that preserve truth are are said to be valid. Or, more generally, we may say that an argument is truth preserving (valid) if and only if the conclusion is a logical consequence of the premises.
  3. The right of publicity was mentioned. The laws regarding right of publicity vary from state to state, but the right of publicity does not seem to pertain in this case. The right of publicity pertains mainly to a likeness being used for publication or public display in connection with commercial products or services. Most ordinarily, a likeness being used in an advertisement and that kind of thing. Taking a photo and then attempting to sell a print of it to the subject of the photo is not a right of publicity matter.
  4. Thanks for that. But, to be clear, my point is not to ask in general about relationships between mathematics and philosophy, but rather to ask the specific question how the incompleteness proof refutes (or conflicts with, whatever) logical positivism or, conversely, what evidence is there of a certain philosophical influence on Godel's proving the incompleteness theorem.
  5. I don't know what you mean by departure from Godel (do you mean that Godel claimed a philosophy of realism regarding infinite sets?). In any case, neither the incompleteness proof nor Godel's proof of it rest on any notion of infinity. As to Cantorian set theory, it is not the notion of infinity that causes paradox. Rather, Cantor himself did not have a formal theory. Later formalizations of it avoid (as far as we know) inconsistency by not including unrestricted comprehension. And I'll give up, for now asking, the unanswered question here as to how a refutation of logical positivism is drawn from the proof of incompleteness.
  6. So what is the argument that the proof of the incompleteness theorem (or the theorem itself) refutes logical positivism or any particular philosophy? As to philosophy influencing math, what philosophical influence do you have in mind regarding the mathematical proof (which - in a certain basic sense - can be formulated within computational arithmetic)?
  7. Asking that someone cite a source is not "moving the goal posts". And you can't justifiably presume that I would dispute the credibility of any source not yet given. I might dispute certain sources but not others (indeed, on certain other points, you've mentioned sources that I did NOT dispute).
  8. Fair enough. So, as I understand now, your view is not necessarily that Godel intended incompleteness to conflict with a certain philosophy, but rather that in fact incompleteness does conflict with that philosophy. So, then you would have to justify that claim. Again, you would then have to show how a philosophical argument is drawn from this particular mathematical proof.
  9. There's no "again" a strawman, since there is no previous strawman, let alone a strawman here. You are welcome to define or redefine your positions. If I have misunderstood your point as being that Godel came up with incompleteness to refute logical positivism or for other philosophical purposes (or even in reaction to logical positivism), then I accept that my best attempt failed to understand whatever it is that you're saying. On the other hand, I don't see how I could be very much faulted for that, for it does seem to be a fair reading of what you actually wrote. So, again, if that is not what you meant, then fine; and meanwhile, my points stand onto themselves (most specifically that we don't have in this thread a citation that connects Godel's proof efforts regarding incompleteness with logical positivism or any philosophy) whether or not in dispute of yours. (Of course, we know that much later in Godel's philosophical development, there may be connections with his mathematical results.)
  10. I have not moved any terms of the question. My replies are exact to each point. And I have not ignored your sources; on the contrary I've addressed them, such as with this one. Next, please cite your source that Godel went on to other questions in logic and mathematics because of his supposed fatigue with misunderstandings of incompleteness (it's possible that is true, but I am curious as to your source). Godel went on to the relative consistency of the axiom of choice and the continuum hypothesis. That was well after he proved incompleteness. And other research he did in the field. It is quite correct to say that the writer of he essay is plainly incorrect on subject. And there is no moving of goal posts by me in that.
  11. "I am under the impression that the particular set of "truths" that are left out are self referential... essentially empty statements with no referent. " [Strictly Logical] That is incorrect. The undecided statements may concern all kinds of matters in arithmetic. The undecided statement (the "G" statement) used to prove the theorem itself may be INTERPRETED as denying its own provability, but the statement itself is a plain statement about natural numbers. It's only by Godel's ingenious work that he hooks up the plain mathematical formula with the matter of unprovability. Again, the statement itself is a basic (though complicated) statement about natural numbers. Moreover, the second incompleteness theorem shows that such statements also evidence that a finitistic consistency proof is not possible. Moreover, by later developments, we find that there is no general solution to Diophantine equations, which is a very basic matter of interest to mathematics. (It's basically to say that there is no algorithm to determine whether an arbitrary equation of the nature as in a high school algebra class has a solution.) And further questions about arithmetic also shown undecidable. It's a lot better to actually investigate the mathematics and context of the incompleteness theorem than to rely on woozy oversimplifications and actual mischaracterizations of it found in many Internet entries.
  12. Whatever the merits of those remarks, they don't at all evidence that Godel devised the incompleteness proof as a way to refute logical positivism or any other philosophy or to advance any particular philosophy. As I mentioned, the incompleteness proof came from Godel's effort to address a certain mathematical problem.
  13. (1) The incompleteness theorem was in 1930 (1931?). That's well before 1946. Godel made major contributions in logic (cf. the Stanford Encyclopedia article) well after 1930. Also, foundations of mathematics includes philosophy regarding mathematics. That later Godel turned a lot of his attention to physics and philosophy does not contradict that he did important work in logic, mathematics, and in foundations of mathematics well after the incompleteness theorem. Moreover, I would need to check on the first publication dates, but Stanford cites papers in logic as late as 1970. (2) I didn't dispute that Godel was not an avid proponent of the ideas of the Vienna Circle. Indeed, it was my point that whatever affinity he might have had for the Vienna Circle, he soon enough moved on from it.
  14. I can't go into the technical details, but in basic terms, the assumptions and logic in the proof can be reduced to nothing more than those of computational arithmetic. Not only is the proof within ordinary mathematics, it's within an even more restrictive criteria: constructive and finitistic. To question the methods of the proof would be tantamount to questioning the methods that make your computer do what it does or even merely those of algorithms for computations on plain counting numbers. As to the explosion principle (a contradiction implies any statement), I don't necessarily want to go into another discussion on it (I've discussed it so many times on forums that it is tedious now), but it too is basic Boolean logic. One can propose non-explosive logics (logicians do study that also), but impugning ordinary mathematics - via certain philosophical or even everyday concerns - really misses the point. As to the paradox of material implication, we can dispense it easily: Let 'P->Q' be merely an abbreviation for '~(P & ~Q)'. I.e. 'if P then Q" (in the specific context of sentential logic, which itself is merely a variation of basic Boolean logic) is merely a way of saying "It is not the case that P is true while Q is false". The use of this in sentential logic is no more than plain Boolean logic. It's what is used for the programs that make your computer do what it does. One can have whatever philosophical objections to such logic, but then it is odd that one doesn't object to it when it makes your computer do what you tell it to do. I mean, when you do a search for results on, say, "NOT(Washington and NOT-Jefferson)" as saying "Any hit you give me on Washington must also be a hit on Jefferson", you don't have philosophical objections to that Boolean logic, right?* *To be fair, ordinarily that search would be an odd one to conduct, since it would bring up every hit that does not even include Washington. But my point is that, while not necessarily of much use, it isn't logically prohibited to want to see only results in which either Washington does not occur or both Washington and Jefferson do occur. A better example: For my business, I want a combined list of all my customers who don't order shovels with all my customers who order both shovels and gloves. That is merely "If S -> G", and has the paradox of material implication with it.
  15. This may help (I'm using 'finitistic' conveniently, not necessarily Godel's or Hilbert's own terminology): Hilbert hoped (indeed, expected) that there would be a finitistic proof of the consistency of analysis. Godel started work to devise such a proof. But in doing that, he realized that instead he could come up with a finitistic proof that if arithmetic is consistent then it is incomplete (perforce that if analysis is consistent then analysis is incomplete) and moreover this provides a finitistic proof that if arithmetic is consistent then there is no finitistic proof of the consistency of arithmetic (perforce that if arithmetic is consistent then there is no finitary proof of the consistency of analysis or set theory), thus that Hilbert's hope (indeed, expectation) was destroyed.
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