tag:blogger.com,1999:blog-23478689.post7698374311052324109..comments2014-08-30T19:38:59.046+00:00Comments on Logic Matters: Gödel Without Tears -- 3Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-23478689.post-89867189290669138012009-10-26T23:36:26.501+00:002009-10-26T23:36:26.501+00:00Thanks for the comments -- I hope the various issu...Thanks for the comments -- I hope the various issues, smaller and larger, and now sorted.Peter Smithhttps://www.blogger.com/profile/03957579588136008664noreply@blogger.comtag:blogger.com,1999:blog-23478689.post-23603012327216845702009-10-26T16:12:54.726+00:002009-10-26T16:12:54.726+00:00Ed, what's leading me into confusion is that s...Ed, what's leading me into confusion is that section 10.1 says "We <b>fix</b> the domain of the quantifiers to be the natural numbers. The result is the language LA."<br /><br />(Not only is the domain fixed, that seems to be part of the definition of the language.)<br /><br />I'm aware that (normally?) there can be nonstandard / unintended models (indeed, one is used in section 10.3); and I can see why an axiom would be needed to rule out models that contain "pseudo-zeros".<br /><br />What's not clear to me is how all that fits with fixing the domain, and moreover to one that doesn't contain pseudo-zeros.a.c.noreply@blogger.comtag:blogger.com,1999:blog-23478689.post-24935875637809137702009-10-26T15:51:43.445+00:002009-10-26T15:51:43.445+00:00To a.c.:
In regards to your query about Section 1...To a.c.:<br /><br />In regards to your query about Section 10, the quantifiers of the language are merely *intended* to range over the natural numbers. You are right that we should not assert from the outset that all L_A-structures have the naturals as their universes, though the text would suggest that reading. In fact, the theory Q itself (like PA) has nonstandard models where the universe is *not* just the natural numbers, but adding in the axiom in question does indeed ensure that Q at least doesn't have models that contain such pseudo-zeroes.Edhttp://www.unwantedcapture.orgnoreply@blogger.comtag:blogger.com,1999:blog-23478689.post-28424903376074865022009-10-26T14:40:38.234+00:002009-10-26T14:40:38.234+00:00Third line of the second paragraph of 10.6: "...Third line of the second paragraph of 10.6: "Q is can capture" should be "Q can capture".<br /><br />Pedantic detail: The first sentence of the 3rd paragraph of 10.6 should say "(and it was isolated by Robinson for that reason)" -- "it" added -- because otherwise it looks like it's saying Q is about the weakest arithmetic that is both sufficiently strong and isolated by Robinson for that reason.a.c.noreply@blogger.comtag:blogger.com,1999:blog-23478689.post-6310312795600921262009-10-26T14:25:52.044+00:002009-10-26T14:25:52.044+00:00Proof of Theorem 12: "We've just shown th...Proof of Theorem 12: "We've just shown that Q |- phi": "|-" should be crossed. That is, whats just been shown is that Q can't prove phi.<br /><br />I can't follow the "in headline terms" part of the proof of Theorem 11. For instance, it says "Adding* S∗n to a yields a", but earlier were were told that x +* a = b. So shouldn't adding* S*n to a yield b?<br /><br />"And adding* S∗a to any x is the same as adding* a (since S∗a = a) i.e. is b" -- but a +* n = a.a.c.noreply@blogger.comtag:blogger.com,1999:blog-23478689.post-57692557548835177412009-10-26T14:02:20.858+00:002009-10-26T14:02:20.858+00:00At first, you use "schemata" (lower case...At first, you use "schemata" (lower case), but once you've introduced some schemata, you always(?) use "Schemata" (capital "S").<br /><br />Perhaps it's capital "S" because "Schema 1", "Scheme 2", etc, would be "S", with "Schemata" as a way to refer to the particular ones collectively; but you don't do the same with "theorems". That is, you use "Theorem 1", "Theorem 2", etc, but not "Theorems".a.c.noreply@blogger.comtag:blogger.com,1999:blog-23478689.post-31348410320296654882009-10-26T13:50:12.159+00:002009-10-26T13:50:12.159+00:00First proof in section 9.3, line 6 says "0 /...First proof in section 9.3, line 6 says "0 /= SS" which is not well-formed. "SS" should be "SS0".<br /><br />A question re section 10: If we fix the domain of the quantifiers to be the natural numbers, why do we have to rule out "pseudo-zeros"?<br /><br />Perhaps I'm missing something obvious, but I could understand it if the aim was to ensure the axioms didn't have models that contained pseudo-zeros; but if we fix the domain to be the natural numbers, ...?a.c.noreply@blogger.com