The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 18 May 2010 02:41 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Nam Nguyen wrote: >>> Marshall wrote: >>>> By your own definition of "true in a theory", this is not >>>> the question. Your definition was "true in all models of >>>> a consistent theory." Since T is not consistent, the >>>> meaning (by *your* definition) of "true in T" is undefined. >>> We live in a binary logic world remember? By _default_, if a formula >>> doesn't meet the definition of being true in a context then it is >>> defined to be false in that context. In technical terms, which >>> I did mention in a conversation with Aatu before, a formula F is >>> true in PA: >>> >>> > F is true <-> (PA isn't inconsistent) and (PA |- F) >>> >>> Just replace PA by my specific inconsistent T and note my >>> "and" above. Do you see now x=x is false in T? >> Btw, that's not my definition as you mentioned above. Although he >> used a slightly different word "valid", the definition could be >> found in Shoenfield's book: >> >> "A formula is valid in T if it is valid in every model of T" >> (Pg. 22) >> >> Note also what I had stipulated above with Aatu and what I said to >> Alan are equivalent. > > It looks like you are claiming Shoenfield's formulation > and yours are *equivalent* (after replacing "valid" with "true"). On pg. 18 (on a "Structure", say, M) he had: "We want to define a formula A to be valid in M if all the meanings of A are true in M". So he defined being "valid" as being "true". > They are not: Shoenfield's version allows a formula to be valid even > for an inconsistent T, and yours does not. Where did he assert or stipulate that?
From: Alan Smaill on 18 May 2010 04:54 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Alan Smaill wrote: >> You make a big distinction between meta-theoretic results >> (which are apparently OK to cite as evidence, eg relationship >> between decidability and truth of purely universal mathematical >> statements); while objecting to statements of basic arithmetic >> properties. In one case, you trust the traditional apparatus, >> in the other you object to just about everything. Why the big difference? > > In a nutshell, it has something to do with the essence of what President > Reagan once said: "trust but verify". In _all_ cases of reasoning through > rules of inference you could verify the agreed correctness of reasoning: > by finite proofs - no trust is required. In some critical cases of reasoning > about certain infinite concepts, such as in basic arithmetic, there's only > trust and no verification is possible. That's not a good reason; there are finite and checkable proofs in Peano Arithmetic also. >>>> Just like reasoning about natural numbers. >>> Reasoning about the naturals numbers on their own merit is actually NOT >>> a logical reasoning. >> >> not *just* logical. >> >>> What is said in Shoenfield's "Mathematical Logic": >>> >>> "The conspicuous feature of mathematics, as opposed to other sciences, >>> is the use of proofs instead of observations." >>> >>> "A mathematician may, on occasions, use observation; for example, he >>> may measure the angles of many triangles and conclude the sum of the >>> angles is always 180 [degree]. However, he will accept this as a law >>> of mathematics only when it has been proved." >> >> Fine. >> >> But still he has axioms for arithmetic; >> where do they come from? In fact, he has more than one axiomatisation. >> Are these random sets of formulas, perhaps? > > The axiom systems for arithmetic come from our intuitive concepts of > arithmetic and they (the systems) are formal (syntactical) descriptions > of the arithmetic concepts, and as such there might be more than one > system describing about the same set of concepts. So no, they're not > random in that sense. OK > But what would all that have to do with the certainty of syntactical > proof as a verification tool and with arithmetic concept as an > intuitive tool and not verification tool, in reasoning? You lay stress on *particular* syntactic proof systems, not just the notion of syntactic proof. You insist that the only true logic (in *your* school) has 2 truth values. Why is *that* an essential feature of mathematical reasoning? >>> The "intuition", the "truth", about the naturals numbers is no more worthy >>> of reasoning than "measure", or "observation" that is mentioned in this >>> book, and which shouldn't be the basis for reasoning. >> >> This applies just as much to the meta-reasoning itself that Shoenfield >> uses thoughout. > > Perhaps some parts of his book, but in general not every meta reasoning is > based on untrusted knowledge of the naturals. It's based on an analysis of the general notion of formal proof, and principles like induction over such proofs (not to mention model-theoretic notions of truth which make use of a chunk of set-theory). How many formal proofs are there? How can I trust any result that claims some property for *all* formal proofs? -- Alan Smaill
From: Alan Smaill on 18 May 2010 05:00 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Nam Nguyen wrote: >>>> Marshall wrote: >>>>> By your own definition of "true in a theory", this is not >>>>> the question. Your definition was "true in all models of >>>>> a consistent theory." Since T is not consistent, the >>>>> meaning (by *your* definition) of "true in T" is undefined. >>>> We live in a binary logic world remember? By _default_, if a formula >>>> doesn't meet the definition of being true in a context then it is >>>> defined to be false in that context. In technical terms, which >>>> I did mention in a conversation with Aatu before, a formula F is >>>> true in PA: >>>> >>>> > F is true <-> (PA isn't inconsistent) and (PA |- F) >>>> >>>> Just replace PA by my specific inconsistent T and note my >>>> "and" above. Do you see now x=x is false in T? >>> Btw, that's not my definition as you mentioned above. Although he >>> used a slightly different word "valid", the definition could be >>> found in Shoenfield's book: >>> >>> "A formula is valid in T if it is valid in every model of T" >>> (Pg. 22) >>> >>> Note also what I had stipulated above with Aatu and what I said to >>> Alan are equivalent. >> >> It looks like you are claiming Shoenfield's formulation >> and yours are *equivalent* (after replacing "valid" with "true"). > > On pg. 18 (on a "Structure", say, M) he had: > > "We want to define a formula A to be valid in M if all the meanings > of A are true in M". > > So he defined being "valid" as being "true". let's leave the terminology aside, and look at the logic. >> They are not: Shoenfield's version allows a formula to be valid even >> for an inconsistent T, and yours does not. > > Where did he assert or stipulate that? When he said: "A formula is valid in T if it is valid in every model of T" How would you express this in FOL? -- Alan Smaill
From: Daryl McCullough on 18 May 2010 06:59 Nam Nguyen says... >> Before you say "wrong" or "right" here, what is your definition of a >> formula being true or false in an inconsistent formal system? I think it's a mistake to use the words "true" or "false" relative to a formal system, when there is already a perfectly good, unambiguous term, "provable". We all know what it means for a formula Phi to be provable in a formal system. Then we can say that Phi is "refutable" if its negation is provable. In general, some formulas may be neither provable nor refutable. "True" and "False" are relative to an interpretation. They are not determined by a formal system. We say that a formula Phi is valid if it is true in every interpretation. What Shoenfield was getting at was that we can also define validity relative to a theory T as follows: Phi is valid in T <-> forall interpretations I, if every formula of T is true in I, then Phi is true in I From this definition, if T is inconsistent, then Phi is valid in T, because the right-hand side of the <-> becomes vacuously true. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 18 May 2010 07:03
Nam Nguyen says... >>> "A formula is valid in T if it is valid in every model of T" >>> (Pg. 22) The distinction between "true in a model" and "valid in a model" is just that the valid statements include statements with free variables that are true in every interpretation of those free variables, while "true" statements must be closed (no free variables). So with distinction, "x=x" is valid, but not true. And it is valid in *every* model. The notion of "valid in theory T" is just this: Phi is valid in T <-> forall models M, if every statement of T is valid in M, then Phi is valid in M If T is inconsistent, then every formula is valid in T. -- Daryl McCullough Ithaca, NY |