Talk:Axiom schema

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 Field: Foundations, logic, and set theory


untitled[edit]

I think this entry needs to reference the inference rule of Uniform Substitution (which doesn't yet have an entry). Uniform subsitution, in sketch: given wff X, in which variable y is free, and another wff Y, we can create a new wff X\[y/Y\] in which every occurrence of y is replaced with Y. 'Uniform' indicates that every y has been replaced with the same wff. The inference rule of Uniform Substitution is: if X is a theorem, and Y a wff, then X\[y/y\] is also a theorem. If Uniform Substitution exists as a given or derivable inference rule in a deductive system, then a great many axiom schemata can be dispensed with (though, given how general the definition in this entry is, not necessarily all). Thoughts?

I have just rewritten this entry, unintentionally doing so without implementing your suggestion. I have no quarrel with what you say, but I do find it very hard to be very precise about substitution rules. I wrote the substitution rules in Laws of Form. I have read that even Alonzo Church himself had trouble getting substitution right in his early formulations of the predicate calculus.132.181.160.42 05:33, 8 August 2006 (UTC)



ELEGANCE of MATHS[edit]


Beauty of mathematics, elegance this is all very very nice. And I am a person who is alive, believe or not, medium or slightly subnormally intelligent than the normal 150 IQ mathematician. And I do like maths, why? Elegance, beauty, God where are we getting? Aesthetics or art pour art or something even higher high in the sky and heavens above us outside the scope of our prying telescopes, where these Deities and Go... dwelllll - no this is not a misprint: this gives the special higher emphasis on the highnesss.. NO!!!

Maths is a very useful and SO nice and lovely tool in your mental arsenal.
Sharp, swift if necessary, slow if we can give us the privilege of having enough time.
This is no ranting, this is a cry of a humble wikipedian to experts who write highly theoretical articles thinking the elegance might get spoiled if we used
PLAIN language, less special symbols, more daily life examples: let us make the mathematical experience come to life through getting down to earth and always when boldly abstracting on things like countability of infinity reconsider whether we
UNDERSTAND OURSELVES and

can make use of our lofty thoughts ONE second later.

Sorry for talking too long and thank you for the delight your artices convey even to those who are not as clever, smart, elegant as you are - there is not irony in these words not at all - caution here please.

--Capekm (talk) 02:52, 1 May 2012 (UTC)

"Loss of 'elegance'"?[edit]

countably infinite[edit]

Article says

Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, ...

I think that is incorrect or needs clarification. I believe it's common in model theory to have languages with uncountably many constant symbols, say one symbol for each real number x. So you could have uncountably many terms to put in place of a free variable. Any restriction against that should be stated explicitly. 69.228.170.24 (talk) 05:31, 24 May 2010 (UTC)