/sci/ - Science & Math

[Anonymous No.6107651]

/sci/, introducing myself to set theory. Trying to understand the Axiom of Comprehension (restricted), and im not following it. I mean I understand the conclusion, that every subset of a set that is defined by a property is itself a set. BUT THE PROOF, I just cant follow it. Here it the proof for those who dont know it.

Consider the property P(x,Q) of x and Q: "x belongs to Q." Then, by the Comprehension Schema (Axiom of comprehension), for every Q and for every P there is a set R such that x belongs to R if and only if x belongs to P and P(x,Q), i.e., if and only if x belongs to P and x belongs to Q. (P plays the role of A, Q is a parameter)

Anyone care to help?

[Anonymous No.6107773]

Which A?

Why is P the name of a proposition and a set?

P(x,Q) is formally not a property, P is. (don't fill in the arugment).

>BUT THE PROOF, I just cant follow it.
Which proof, I think that's just the definition of the schema - or what do you try to do?

[Anonymous No.6107935]

>>6107651
>introducing myself to set theory
Don't.
>Trying to understand the Axiom of Comprehension
You have to be like understand shitloads of mathematics to decide whether it is descriptive to mainstream math or not, and yea descriptive, so it is masturbation and it is not the right thing. That would likely be the best you can do about it.
Therefore, you don't try understand ZFC by "studying" ZFC, you understand math to understand how ZFC was laid and maybe study them lazily(as needed).
see Wittgenstein's note on "Remarks on the Foundations of Mathematics"
http://www.hse.ru/data/366/347/1234/The%20Collected%20Works%20Of%20Ludwig%20Wittgenstein.pdf

There are people who follow similar spirit of Bourbaki, which derps about "rigor" but is in fact Axiomatics, so not at all rigorous.

I found this interesting paper which basically said right out on what I think about certain individuals on 20-th century math.
https://www.dpmms.cam.ac.uk/~ardm/bourbaki.pdf

[Anonymous No.6109141]

>>6107773
Honestly im not fluent enough with set theory to help you, im honestly looking for help. However functions arent defined in the usual way, and they have their own set of symbols to go along with it.

It is a proof of a schema, the schema itself (not with the symbols) would be "...every subset of a set that is defined by a property is itself a set."

To best explain whats going on in the proof (god or /sci/ help me im going to bastardize this)

There is a property, x belongs to Q (belongs to here has the symbol of a horseshoe with open end facing the right), symbolized as, P(x,Q) of x and Q.

The rest follows with setting up what belongs to what (which is the member of the set and which is the set, which then, supposedly, shows that this is a set of sets, since subsets now become sets themselves).

>>6107935
Im taking a class on set theory, im not going in blind.

Would you mind telling me in your own words why you cant study the foundation before doing the math with assumptions, only later to question the assumptions?

Ill read what youve posted, but if this just leads to Wittgenstein's ladder then I've already heard about it

[Anonymous No.6109150]

>>6109141
me again, just read the second article and though I should mention that

>>6107935
In my set theory series we deal with Russel, ZFC, and incompleteness.

[Anonymous No.6109387]

>>6109141
>Im taking a class on set theory, im not going in blind.
No, you are in blind, there is a camp of people(the origin is probably Bourbaki) who wish to setup this course badly under the name of "rigor". That's why it was said to be a disease.
>Would you mind telling me in your own words why you cant study the foundation before doing the math with assumptions, only later to question the assumptions?
>study the foundation
Because ZFC and thousands alternatives out there are all not the foundation. They have messed up its name, there is no foundation in it. They are at best description and may well be wrong.
>only later to question the assumptions?
Technically, you can. It is not like I try to stop you.
If you understand the misinformation first without knowing anything about it, then it limits your scope of thinking and it is hard to adjust oneself and relearn/unlearn stuff. At least that's how I see it.
>Ill read what youve posted, but if this just leads to Wittgenstein's ladder then I've already heard about it
I think you mean his "Philosophical Investigation" but he was also Russell's student.

[Anonymous No.6110353]

>>6109387
Why are they only "description", what exactly do you mean by this?

[Anonymous No.6110381]

>half-decent maths thread on /sci/
>its fucking set theory again
Thank you for nothing.

[Anonymous No.6110428]

>>6110381
Wait how was it half decent but then also worth nothing when the whole thread has been about the same thing?