fom
Posts:
1,968
Registered:
12/4/12


Re: Ordinals describable by a finite string of symbols
Posted:
Jun 30, 2013 3:08 PM


On 6/30/2013 1:04 PM, apoorv wrote: > On Sunday, June 30, 2013 9:13:25 PM UTC+5:30, fom wrote: >> On 6/30/2013 9:05 AM, apoorv wrote: >> >> >> >>> Well, my sense is that the infinite enters mathematics or logic through the universal >> >>> Sentence..The universal sentence attempts to use ' a finite string ' to represent or define >> >>> Something infinite. To me, it is an appealing proposition, that 'Infinity cannot be finitely represented'. I made some posts earlier along those lines, but obviously, the mainstream >> >>> View is different. >> >>> Apoorv >> >> >> >> I saw those posts and had been appreciative of >> >> the remarks. >> >> >> >> You are basically correct. > Thanks for that remark, Though what followed is far from clear to me.
That is because "innovators" often innovate by disposing of history. The logic you are about to invoke did not spring forth as the goddess Athena from Greek mythology.
> The universal statement in a finite domain of discourse is nothing but a wff > (conjunction)Of sentential logic.
If you are presupposing a finite domain of discourse, how could your logic fail to give you the result which it did?
Along a different line of thought, you are interpreting the universal quantifier. There are other interpretations.
For example, the words "arbitrary substitution" comes to mind. This would be relevant with respect to an indeterminate plurality as opposed to a definite totality. The distinction is very clear in Markov's "Theory of Algorithms" where the domain of discourse is restricted to finitely constructed sequences of symbols taken from finite alphabets. The reinterpretation of the quantifier holds that "given an object satisfying the constraint of the domain..." It is not presupposing that the domain is a totality. The constraint may be verified recursively, and, the quantifier is only applied to objects of verified origin. There is no interpretation as an infinite conjunction.
> Sentential logic does not allow us the luxury > Of an infinite conjunction. The universal statement, with an infinite domain of discourse, > Allows us to achieve the same .
... when interpreted as you are interpreting it.
Maybe.
> Put differently, every wff of sentential logic can be mapped to a Boolean function from > The domain {0,1}^n to {0,1}.FOL , with it's Universal statement, allows > Us wff s with the domain {0,1}^00 and associate truth values with essentially infinite > Ordered Strings like (1,1,1...).
Yes. It was one of my first independent questions as an undergraduate. "How is compactness in logic related to compactness in topology?"
Whoops! What is the dichotomy in topology?
Finite vs Arbitrary
> The probability of AxPx being True, , in the context of infinite domain and independent P(x) s > Is Zero.Equivalently, the information content is infinite.
Yes. As I looked into one of the authors that David Petry brought to the attention of the group, it is clear that there are close relationships with probablistic methods and logic.
I believe, however, that one might argue that there is a difference between "all" and "all except on a set of measure 0". This, apparently, is a criticism of Freiling's axiom in set theory.

