Stack Exchange
log in

user709833

 Mathematics

Associated with Math.SE; for both general discussion & math qu...
4
user709833
Oct 3, 2019 13:08
can anyone help me understand better when we would need second order logic as opposed to getting by with first order
user709833
Oct 2, 2019 14:00
since we have these second order Peano axioms as well as these first order Peano Arithmetic axioms
user709833
Oct 2, 2019 13:59
I am a little confused what most of our modern mathematics is "running on"
user709833
Oct 2, 2019 13:59
@AlessandroCodenotti For instance if you wrote a proof that at some point used real numbers or even something simple like distribution law

And i say wait how does that work, how do you know / prove that works like you say it does

You would go to the second order peano axioms and their resultant proofs/derivations/etc?
user709833
Oct 1, 2019 14:38
is this correct?
user709833
Oct 1, 2019 14:38
and from the set theory build functions?
user709833
Oct 1, 2019 14:37
and then from there you use FOL to build set theory and other first-order theories like PA?
user709833
Oct 1, 2019 14:37
like you start with propositional logic, and from there build FOL
user709833
Oct 1, 2019 14:37
sort of the bird's eye view of how everything is built
user709833
Oct 1, 2019 14:37
@AlessandroCodenotti that's basically what I am trying to get at
user709833
Oct 1, 2019 13:33
@AlessandroCodenotti My question though is that if we are using functions does it mean that we must also be using a set theory underlying it first / have one defined?
user709833
Oct 1, 2019 02:15
is there some formal definition in set theory?
user709833
Oct 1, 2019 02:15
what is what i am asking
user709833
Oct 1, 2019 02:10
like set behavior is defined with all this axiom stuff but what about functions
user709833
Oct 1, 2019 02:10
like what *are functions exactly
user709833
Oct 1, 2019 01:59
ignoring category theory stuff, do functions typically require some kind of set theory in order to use?
 

♦  Logic

This room is meant for discussion about logic, including found...
1
user709833
Oct 2, 2019 15:08
Sorry I am confused, before you said "It turns out that almost every number theoretic fact can be proven in (first-order) PA" and then you said "It is fair to say that PA cannot capture most number theory"
user709833
Oct 2, 2019 14:59
is second-order logic required at all for establishing the reals? @user21820
user709833
Oct 2, 2019 14:26
the second order peano axioms or the first order PA system?
user709833
Oct 2, 2019 14:25
what would you resort to?
user709833
Oct 2, 2019 14:25
like say you write a proof of some random number theory or calculus concept and partway in you have a line that uses a(b + c) = ab + ac and I go "huh that's kinda peculiar, what is the proof that that works"
user709833
Oct 2, 2019 14:25
does our usual mathematics rely on the first or second order system?
user709833
Oct 2, 2019 14:12
is a theory just a set of axioms/schemas?
user709833
Oct 2, 2019 14:11
or the "language"?
user709833
Oct 2, 2019 14:10
that is another word I've seen a lot
user709833
Oct 2, 2019 14:10
when you say PA only uses 0 and 1, is this part of the "signature"?
user709833
Oct 2, 2019 14:10
right, yeah, I think tao only relies on 0 at first but then says we can just define S(0) = 1, S(1) = 2, S(2) = 3, etc
user709833
Oct 2, 2019 14:08
like x * 0 = 0 or x * 1 = x?
user709833
Oct 2, 2019 14:08
why does PA only have 0 and 1? minimal number of constant symbols to describe the basic axioms?
user709833
Oct 2, 2019 14:06
is a "schema" something where we can write infinitely many axioms where each one literally looks different?
user709833
Oct 2, 2019 14:04
are you referring to something like how we're really only saying things like S(S(S(0))) instead of 3
user709833
Oct 2, 2019 14:02
I was reading a book called Analysis Volume I (by Terrence Tao) where he uses PA to derive the natural number laws and build up to integers, rationals, reals etc
user709833
Oct 2, 2019 13:57
so I don't understand the difference between "an axiom with infinitely many inputs" and "an axiom schema with infinitely many axioms you can make"
user709833
Oct 2, 2019 13:57
When people say a schema is infinitely many axioms I get sort of confused by that because if I have the axiom x = y iff S(x) = S(y), to me that also seems kind of infinite because I can pick infinitely many natural numbers for either x or y
user709833
Oct 2, 2019 13:55
@user21820 I am still a little confused though if only because I feel like I can write a program for just about anything I want
user709833
Oct 1, 2019 23:08
like if we just listed that statement by itself would it be clearly a schema?
user709833
Oct 1, 2019 23:07
If we have an axiom schema do we need to literally describe it like that? as an axiom schema metamathematically? and say that "for every formula" etc?
user709833
Oct 1, 2019 22:34
i googled it but hard to understand, it still looks the same
user709833
Oct 1, 2019 22:34
i don't really understand the difference between axiom and axiom schema and single formula, they all look the same to me
user709833
Oct 1, 2019 22:34
metamathematically describe the schema? what do you mean by that?
user709833
Oct 1, 2019 22:26
how so
user709833
Oct 1, 2019 22:07
like what is the correct way to write induction
user709833
Oct 1, 2019 22:07
@MaliceVidrine Do you know how induction is presented logically/formally?
 

 Math Meta Chat

Chat-room for Math Meta stuff (for moderator-related stuff go to
1
user709833
Oct 2, 2019 00:09
wow that sounds intense
user709833
Oct 2, 2019 00:07
no one seemed to have any actual details on why the demotion took place
user709833
Oct 2, 2019 00:07
i tried reading through the threads but it was a little confusing
user709833
Oct 1, 2019 23:59
whats the primary controversy exactly? the demoting of the mods? @XanderHenderson
user709833
Oct 1, 2019 22:15
why are people upset about the switch @XanderHenderson
user709833
Oct 1, 2019 22:14
whats so bad about the new license?
 

 Discussion between Noah Schweber and …

Imported from a comment discussion on math.stackexchange.com/q...
user709833
Oct 1, 2019 05:24
Do you know of anything that is accessible / aimed at beginners, but not skipping over crucial details?