Re: axioms of mathematical logic


itaj wrote:

i'm a math student on third year. i took many courses in analysis and
algebra but there won't be a logic course this year in my university.
so i decided to learn logic from a book. now i have a problem/question,
but i'm not sure how to phrase it exactly. so i'll have to give some
background.

the way i learned math until now was:
first was taught the language of predicate logic, and the basic rules
to manipulate expressions (i think they're called inference rules).
then given the list of ZFC axioms.
then we built everything in that setting (all analysis and algebra i
learned). all objects i dealt with until now were sets defined with
ZFC.
in set theory course the ordinals and cardinals were defined too, all
based on ZFC. also proved the transfinite induction and zorn's lemma.

and for me it's really fine. i like math to be formal just like that.

now i started learning logic and in the book they talk about
expressions as if they are themselves some objects they can manipulate.
they chose a language and define the set of all expressions over the
language.
but then saying they can chose the axioms to use only after that.
so my problem is more or less: how can the '''set''' of expression over
a language be a '''set''' if i haven't yet chosen the axioms (be ZFC or
others) to define it as a set?
also like the valuation functions. a function f:A->B must be a subset
of AxB (or another set in some equivalent way, but a '''set'''
nontheless) therefor a '''set''' by ZFC.
by what axioms the set of constant letters in the language is a
'''set'''?
in what sense a model is a '''set'''?
is it all done already inside the setting of ZFC or some other list of
axioms?
they use zorn's lemma on sets of sentences to prove a compactness
theorem.
if so, can it be done under other settings too?

if it's long or complicated at least direct me to where i can find
answers.

1. At the start of sentences, and paragraphs, we use capital letters.
2. While the letter "i" can be used in lower-case in words, the
personal pronoun 'I' is always written in its capital form as 'I'.

.

Relevant Pages

  • Re: axioms of mathematical logic
    ... i base it on predicate logic and ZFC. ... manipulate expressions and objects without first setting the rules. ... first was taught the language of predicate logic, ... but then saying they can chose the axioms to use only after that. ...
    (sci.logic)
  • Re: axioms of mathematical logic
    ... first was taught the language of predicate logic, ... then given the list of ZFC axioms. ... expressions as if they are themselves some objects they can manipulate. ... but then saying they can chose the axioms to use only after that. ...
    (sci.logic)
  • Re: axioms of mathematical logic
    ... first was taught the language of predicate logic, ... then given the list of ZFC axioms. ... expressions as if they are themselves some objects they can manipulate. ... but then saying they can chose the axioms to use only after that. ...
    (sci.logic)
  • Re: axioms of mathematical logic
    ... first was taught the language of predicate logic, ... then given the list of ZFC axioms. ... expressions as if they are themselves some objects they can manipulate. ... but then saying they can chose the axioms to use only after that. ...
    (sci.logic)
  • Re: axioms of mathematical logic
    ... that's also what real dictionaries do:) ... first was taught the language of predicate logic, ... then given the list of ZFC axioms. ... but then saying they can chose the axioms to use only after that. ...
    (sci.logic)