Re: axioms of mathematical logic


well, i'm not sure i understood exactly what you said here.

did you mean that something called "generalized theory of primitive
recursive functions only" gives a formal begining to mathematical
logic?

and then that there's intuitionism which is...?

and then conceptual backing. well i'm not sure if you mean something
that reasons math and logic into the real world, which currently less
interests me.

i know about the pitagorians problem with incompleness of rational
field. if this is what you ment. and also the way real numbers can be
build starting with ZFC. i have no questions about the base of
mathematics. i base it on predicate logic and ZFC.
my problem here is only when i start studying mathematical logic, which
seems (in the book i have and some other references on the internet) to
manipulate expressions and objects without first setting the rules. now
this book does this generously. that is, putting things into sets and
functions between them. then using nonthing less than zorn's lemma to
prove somthing called "compactness theorem". that looks like assuming
ZFC first, but they didn't even mention it. _and this can't go without
saying_, not for me.

i don't know about "analytica priora". i suppose that 2000 years ago
time was too early for what i'm looking for. they didn't know yet about
the "russel's set paradox" and incompleteness theorems of goedel. and
for all i know they didn't build anything that can deal with these
issues. fix me if i'm wrong.

the way i see it (math and mathematical logic), is a game. what i'm
looking for is a clear explenations of the rules. but right well from
the start, with no shortcuts.

itaj

tohentoon wrote:
In substance I sympathize with much of smn's exposition,

but

while reference to a (generalized, not arithmetical exemplified ) theory of
'primitive (!) recursive functions only' might do the explicandum job very
well on a highly formalized level ( there are texts in a context of studying
intuitionism, I remember vaguely, of Georg Kreisel, which are important in
this respect )

imop

a first access for a conceptual backing of formalized theories like FOL,
ZF(C), Peanos |N, or what You prefer as Your example
will learn a lot of reflecting simply, what conceptual situation occured as
well with ancient pythagoreans operations on natural else rational numbers
or with the imperfect and in some basic sense finite symbolism of
'analytica priora'

Don't mistake these early accounts for incomplete implementations of modern
formalized theories, these theories are especially in so far of a special
kind, as they are n o t themselves arithmetized ( Gottlob Frege did not
teach in the academy nor at any other ancient spot )

Hope, these references not too vague to communicate my suggestion

Regards


"smn" <smnewberger@xxxxxxxxxxx> schrieb im Newsbeitrag
news:1164404886.040407.161040@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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.

This is a very nice question which is ignored or glossed over in
textbooks.For most of what you are talking about (like using Zorn's
Lemma to prove compactness theorems ) the short answer is that the
language or type of languages studied (say laguages using first order
predicate logic) is being modeled within the set theory that you set up
to do analysis ,geometry ,algebra ,etc .The alphabet and other basic
expresions are modeled by a set in ZFC and statements , are defined to
be finite sequences (ie n-tuples,or functions on{1,2,3...n} .The
sequences are interpreted as actual sentences of a ,say writtten
language in the same way that 3-tuples are interpreted as points of
physical space in a course in mechaniics.
However if the language you are really interested in is the one you
set up in your studies and mentioned in your initial remarks there is
some problems of circularity which is probably behind that is
bothereing you as it did me.For the elementary part of such a study
you don't need anything near as strong as ZFC ,You need the integers
and the axioms needed for the definition of a function as a set of
ordered pairs and you need mathematical induction,and more particularly
recursive methods.These are referred to by logicians as finitary
methods.I don't know a good reference but the logicains are familiar
with your question .The book on Set Theory(3rd edition) by the logician
Robert Vaught gets into consistancy proofs in the later chapters and
provides a partial answer.Perhaps a better informed Logician will
respond with some refierences for doing the foundations for studying
the foundations of analysis.Regards,smn


.

Relevant Pages

  • 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
    ... 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)
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