Re: The shocking truth about the naturals

george wrote:
On Sep 6, 9:31 pm, Boris Borcic <bbor...@xxxxxxxxx> wrote:

ng as this old argument is, it is curious it has
escaped the determined skeptic that the upward Löwenheim-Skolem
theorem incontrovertibly shows we can't really prove any collection
countable.

Since the above technically alludes to me (as current last representative), I
guess it is legitimate for me to remark on it... that (1) my /own/ use of
"incontrovertibly" indeed alluded to the unavoidability of "determined skeptics"
and further adepts of the church of univocity, and (2) that my alluding to the
necessity of counting with them is not sufficient cause to count myself among
these groups.

I am a little surprised that AK didn't give you credit for bringing
this up.
It is true that it is an intellectual point in its own right,
regardless of who
said it last, but since I wasn't reading the "physical process"
thread, I
didn't realize that you had inspired him.

Well, thanks for caring, the real point for me is not one of priority but one of community, in the sense that it was an issue I spotted silently some 25 years ago while browsing on my own through the "fundations" and "set theory" shelves of some math library. A confirmation that the issue wasn't just a private invention was deserved, and AK's reluctance to provide me with any... was none too welcome. Your own intervention, when I finally asked Google about "Wittgenstein+Loewenheim+Skolem" to discover this thread by way of the sci.logic archives, entailed the nice discovery that my assessment was correct.


The relevant point here (which is converging with the point of my long
thing
from Sept.6) is that the situations are INHERENTLY not symmetric.
"Prove" does not have just one meaning. Around here, we are
artificially
RESTRICTING the meaning of prove to "prove in classical finitary 1st-
order logic".
This paradigm is founded on the concept of a FIRST-ORDER LANGUAGE.
This language (or method of language-formation, founded on simple
"signatures") HAS A *DEFINITION*.


I think I understand that. Proofs are relative to a theoretical frame. And languages having definitions is not a mystery for an applications programmer. I'll re-read your sept.6 piece with interest. I am not sure I understand where you locate the dissymmetry, because I don't understand (yet) how the restriction to "prove in classical finitary 1st-order logic" creates a(n artifical) symmetry (what you appear to be saying). Instinctively it appears clear that the downward theorem impacts on interpreting the upward theorem in a way that has no converse (but, well, instincts...).

And I do not quite understand how the proof for the upward theorem (as sketched in the wikipedia) could be said to cather to "classical finitary 1st order logic" just as well as does the downward proof (since the former enrolls non-countably many first-order predications of differences between non-countably many constants; or else it is the case that I don't understand the intent of "finitary" - not to speak of the proof itself).

Notwithstanding the above; and while the downward and upward proofs (wikipedia sketches) do look quite different, I am wondering - more as an artistic exercize than anything else - how far could succeed an effort to formulate a simultaneous proof of both the upward and the downward theorem(s), given that their statements show lots of common structure.

Regards, Boris Borcic


.

Relevant Pages

  • Re: The shocking truth about the naturals
    ... escaped the determined skeptic that the upward Löwenheim-Skolem ... Since the above technically alludes to me, ... "Prove" does not have just one meaning. ... This paradigm is founded on the concept of a FIRST-ORDER LANGUAGE. ...
    (sci.logic)
  • Re: PING: Jerry F.
    ... Sometimes the freeway is above the crossroad, ... pretty fierce upward slope.. ... downward so that the entry will be downward. ... Oh, you are making sense, but you must have very humpy freeways. ...
    (alt.usage.english)
  • Re: Ascending/Descending icons
    ... there that resembles notions of "upward" or "downward". ... Maybe I'm in a locale that doesn't parse string representations of Unicode characters into those Unicode characters. ...
    (comp.lang.java.gui)
  • Re: Possible energy invention by James M Driscoll aka Spaceman?
    ... causing the floatable masses to now move downward and upward again creating ... electricity again. ... As I suspected simply another 'perpetual motion' device. ...
    (sci.energy)
  • =?ISO-8859-1?Q?Re:_Has_BS_Anand_Refuted_G=F6dels_Theorem?=
    ... The downward is the one that is important. ... does NOT contradict that one, since the standard model is ... the complete irrelevance of the upward one. ...
    (sci.logic)