'When Are Relations Neither True Nor False?
in [Math]
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From: Jesse F. Hughes on 2 Apr 2010 13:55 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Daryl McCullough wrote: >> Newberry says... >> >>> What I was getting at is how we know that the system is consistent. >> >> In the case of PA, it's because we know that everything it says >> about the natural numbers is true. > > What is the natural numbers collectively? One in which "There are > infinitely many examples of GC" is true? Or one in which "There are > infinitely many counter examples of GC" is true? Are you sure you > know what you're talking about, in talking about the "natural > numbers"? You do not have to know everything about N in order to know what N is. I don't know whether the moon has hundreds of craters 100 yards across or thousands. Or dozens. Yet, I'm sure I know what the moon is. And I also know certain things about the moon. -- Jesse F. Hughes "You may not realize it but THOUSANDS of people read my posts. You are putting your stupidity on wide display." -- James S. Harris knows about wide displays of stupidity.
From: Nam Nguyen on 2 Apr 2010 14:24 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Daryl McCullough wrote: >>> Newberry says... >>> >>>> What I was getting at is how we know that the system is consistent. >>> In the case of PA, it's because we know that everything it says >>> about the natural numbers is true. >> What is the natural numbers collectively? One in which "There are >> infinitely many examples of GC" is true? Or one in which "There are >> infinitely many counter examples of GC" is true? Are you sure you >> know what you're talking about, in talking about the "natural >> numbers"? > > You do not have to know everything about N in order to know what N is. > > I don't know whether the moon has hundreds of craters 100 yards across > or thousands. Or dozens. Yet, I'm sure I know what the moon is. And > I also know certain things about the moon. > Right. We'd have an _incomplete_ knowledge about the moon, in this example. That would enable us to know, say, there's at least 1 crater on its surface. But that wouldn't allow us to know truth of one simple quantification predicate statement "All the moon craters are at least 100 yard across". The issue here is not "what N is" per se. One could easily _define_ N as a model of some modulo arithmetic. The issue is how we can know the _syntactical consistency_ of PA, using strictly definition of syntactical consistency. Just defining "things" that we don't completely know does not _prove_ in meta level that PA is syntactically consistent. We shouldn't invent something that isn't compatible with rules of inference, and then "prove" things that the rules themselves can *not* prove.
From: Nam Nguyen on 2 Apr 2010 20:32 Nam Nguyen wrote: > Jesse F. Hughes wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Daryl McCullough wrote: >>>> Newberry says... >>>> >>>>> What I was getting at is how we know that the system is consistent. >>>> In the case of PA, it's because we know that everything it says >>>> about the natural numbers is true. >>> What is the natural numbers collectively? One in which "There are >>> infinitely many examples of GC" is true? Or one in which "There are >>> infinitely many counter examples of GC" is true? Are you sure you >>> know what you're talking about, in talking about the "natural >>> numbers"? >> >> You do not have to know everything about N in order to know what N is. >> >> I don't know whether the moon has hundreds of craters 100 yards across >> or thousands. Or dozens. Yet, I'm sure I know what the moon is. And >> I also know certain things about the moon. >> > > Right. We'd have an _incomplete_ knowledge about the moon, in this example. > That would enable us to know, say, there's at least 1 crater on its > surface. > But that wouldn't allow us to know truth of one simple quantification > predicate statement "All the moon craters are at least 100 yard across". > > The issue here is not "what N is" per se. One could easily _define_ N as > a model of some modulo arithmetic. The issue is how we can know the > _syntactical consistency_ of PA, using strictly definition of syntactical > consistency. > > Just defining "things" that we don't completely know does not _prove_ > in meta level that PA is syntactically consistent. > > We shouldn't invent something that isn't compatible with rules of > inference, and then "prove" things that the rules themselves > can *not* prove. Let me put to rest the idea we know enough about the natural numbers, to prove important thing such as the consistency of PA. I'll do that by pointing out the existence of a specific unknown natural number. Let N be the set of natural numbers and R the set of standard reals. Let a natural number n be expressed as n = d0d1d2...dn, where d's are the decimal digits. Let's also define the following functions: f1: N -> N, f1(n=d0d1d2...dn) = dn...d2d1d0 f2: N -> N, pE(n) = p, where p is the greatest prime <= the even n [assuming n >= 0]. Let S1 = {n | n is an example of GC} Let S2 = {n' | n' is a counter example of GC} Note that at least one of S1, S2 must be infinite. Now if S1 is finite of length l > 0, then there is an infinite sequence: Seq1: n1, n2, ..., nl, 0, 0, 0, ... where all terms are either in S1 or 0. (If S1 were empty, then all terms are defined equal to 0). Similarly, an _infinite_ sequence Seq2 would exist, where all terms are either in S2 or defined to be 0. Let's define the set S as: Let S = { m | max(f1(f2(nth-term-of-Seq2)),f1(f2(nth-term-of-Seq1))) } By Well Ordering Principle, S has a minimal number which would be the called Un: the desired "unknown" natural number. To know the natural numbers then is to know the value of Un, which we can not know.
From: Nam Nguyen on 2 Apr 2010 20:34 Nam Nguyen wrote: > > Let me put to rest the idea we know enough about the natural numbers, > to prove important thing such as the consistency of PA. I'll do that > by pointing out the existence of a specific unknown natural number. > > Let N be the set of natural numbers and R the set of standard reals. Please disregard my mentioning about the reals (R) here. Thanks. > Let a natural number n be expressed as n = d0d1d2...dn, where d's are > the decimal digits. Let's also define the following functions: > > f1: N -> N, f1(n=d0d1d2...dn) = dn...d2d1d0 > f2: N -> N, pE(n) = p, where p is the greatest prime <= the even n > [assuming n >= 0]. > > Let S1 = {n | n is an example of GC} > Let S2 = {n' | n' is a counter example of GC} > > Note that at least one of S1, S2 must be infinite. Now if S1 is > finite of length l > 0, then there is an infinite sequence: > > Seq1: n1, n2, ..., nl, 0, 0, 0, ... > > where all terms are either in S1 or 0. (If S1 were empty, then all terms > are defined equal to 0). Similarly, an _infinite_ sequence Seq2 would > exist, where all terms are either in S2 or defined to be 0. > > Let's define the set S as: > > Let S = { m | max(f1(f2(nth-term-of-Seq2)),f1(f2(nth-term-of-Seq1))) } > > By Well Ordering Principle, S has a minimal number which would be > the called Un: the desired "unknown" natural number. > > To know the natural numbers then is to know the value of Un, which we > can not know.
From: Transfer Principle on 2 Apr 2010 21:38
On Apr 1, 5:30 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Aatu Koskensilta wrote: > > How do I know that Peano arithmetic is consistent? I know it the way I > > know any mathematical theorem I have personally proved. > So what you're saying is you just _intuit_ PA system be consistent, > no more no less. Of course anyone else could intuit the other way too! I agree wholeheartedly. One poster's intuition is another poster's counterintuition, and just because the standard theorists believe in all the theorems of PA that they "have personally proved," it doesn't mean that all posters must share in that belief. To repeat, what may be intuitive to one poster may be counterintuitive to another. And I see no reason to favor one poster's intution over another's, no matter what the standard theorists try to say. |