The End of an Era - That of The Natural Numbers
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From: David R Tribble on 22 Jun 2010 18:28 Nam Nguyen wrote: > The important consequence of the above observation is [...] > We're not quite there yet, but the above paragraph has brought us closer > to the title of the thread, since it marks the beginning of an end of > the absoluteness of the truths of the natural numbers. I've read that last sentence about a dozen times now, and I still can't make heads or tails of it.
From: Nam Nguyen on 22 Jun 2010 23:44 David R Tribble wrote: > Nam Nguyen wrote: >> The important consequence of the above observation is [...] >> We're not quite there yet, but the above paragraph has brought us closer >> to the title of the thread, since it marks the beginning of an end of >> the absoluteness of the truths of the natural numbers. > > I've read that last sentence about a dozen times now, and I > still can't make heads or tails of it. I'm not surprised given most of the thread has been devoted to that one issue on the relativity of x=x being true, within FOL=. Let me first use examples from the familiar SR to illustrate some principle of relativity in SR, then I'll focus back to the mathematical case of the natural numbers. Now, Let's define the following: F = the _standard_ frame of reference, which is one of the 2 frames each of which is associated with one car out of 2 cars, moving toward each other with a non-zero constant speed. S1 = "Events E1 and E2 happened simultaneously." S2 = "The speed of light is 300,000.0 km/s." As far as SR is concerned, F is _not uniquely defined_ hence the truth of S1 is _relative_. If S1 is true in F it's false in the other one, which could also equally be designated as F. On the other hand, the truth of S2 is _absolute_ since it's true in F as well as in the other frame (as well as in all frames of reference). *** In (FOL) mathematics similar relativity situations would exist. For instance, suppose L(0,+) be a language to describe basic group theory, and suppose we've _partially_ spelled out a model M as: U = {{}, {{}}, ...} M = {<'A',U>, <'0',{}>, <'+', P = {<{},{},{}>, <{{}},{{}},{}>, ...}>} Where the first "..." means U would have _at least_ one more (but could be more than 2) elements. From the predicate P, we know this formula is true: (a) Ex[x+x=0] [I.e. x is the inverse of itself.] However the formula: (b) Exy[~(x=y) -> (x+y=0)] is indeterminable in M, simply because we don't know how many more elements in U "..." would mean. In this case, M is very much akin to the the frame of reference F in the SR example above: like F, M is _not uniquely defined_. And consequently the truth of (b) is relative in M in the sense that there are other model _with the same description_ that would have opposite truth value for (b) than in M. *** The case for relativity of (FOL) natural truths is very much similar to F or M above. We'll demonstrate that the so call the standard model of L(PA), denoted by N, which is purported to be the natural numbers, collectively speaking, can't be uniquely defined using model definition. Hence certain truths such as (1) [previously defined] isn't absolutely true in N, just like S1 in F or (b) in M above. [To be continued...]
From: Aatu Koskensilta on 23 Jun 2010 08:53
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > The question I have for you, Aatu, and Marshall though is why you > three think you're infallible and above the rigorousness of > mathematical reasoning? Why do you think I think I'm infallible and "above the rigorouness of mathematical reasoning"? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |