Re: axioms of mathematical logic

On 24 Nov 2006 11:39:28 -0800, itaj <itajsherman@xxxxxxxxx> said:
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.

And indeed they are.

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.

Well, no. The axioms were chosen before there were well-defined formal
languages. The formal languages simply provided a medium in which the
axioms could be expressed with great clarity and precision. For
example, one of Zermelo's original axioms (his "Axiom of Separation")
was that, for any given set S and any "definite property" P, there is a
set consisting of exactly those elements of S that have property P. The
idea of a "definite property", however, being expressed in ordinary
language, is quite vague and, consequently, subject to a number of very
different (and, if similarly informal, similarly vague) interpretations.
With rigorous formal languages at our disposal, we can express various
alternative interpretations of the original informal idea very clearly
and and precisely. One can then simply choose the language appropriate
to the interpretation that one wants. Additionally, using formal
languages enables us to prove much deeper results about, e.g., the
connections between a theory like ZFC and its models.

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?

Sets didn't come to be in virtue of some choice of axioms. There were
sets before there were formal theories of sets, just as there were
numbers before there were any formal number theories and masses and
forces before Newton. Set theories just express very clearly the basic
principles that we believe govern the nature of sets, just as the axioms
of Peano Arithmetic clearly express the basic principles we believe
govern the nature of the natural numbers and as Newton's laws express
the basic principles that govern the motion of, and interactions
between, physical objects (of appropriate mass and velocity).

.

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)