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18m
accepted Can we have $x\in A$ and $x\in A\times B$?
2d
comment Set-builder Notation
I would translate $A=\left\{x:\phi (x)\right\}$ as $\forall x:[x\in A\iff \phi(x)]$.
Jul
22
comment Books for Ordinals and Cardinals
I'm guessing that the first few chapters or an appendix of your textbook with cover all the set theory you will need without going needlessly down the rabbit hole of ordinals, cardinals and ZFC.
Jul
21
comment Omitting parantheses in formulas
Binary operators usually have precedence from left to right. Occasional exception: ^ operator for exponentiation. Usually 2^3^4 $\equiv$ (2^3)^4. Sometimes 2^3^4 $\equiv$ 2^(3^4). If exponentiation indicated by superscripts, then precedence is from right to left, e.g. $2^{3^4}\equiv 2^{(3^4)}$.
Jul
21
comment Omitting parantheses in formulas
The $\neg$ operator (and prefix operators in general) have precedence for right to left, e.g. $\neg\neg\neg P \equiv \neg(\neg(\neg P))$.
Jul
18
comment Logical Implication & Injective Functions
You are forgetting the key condition that makes the implication false, namely (in your non-standard notation) if (( A == true) and (B == false)). Otherwise, the implication is true.
Jul
17
asked Can we have $x\in A$ and $x\in A\times B$?
Jul
17
answered Logical Implication & Injective Functions
Jul
17
revised Theory of real numbers and using functions
deleted 16 characters in body
Jul
17
revised Theory of real numbers and using functions
added 51 characters in body
Jul
17
comment Theory of real numbers and using functions
You don't need ZFC to study real analysis. Most introductory texts will give you all the set theory you will need including the theory of functions in the first chapters, and in a much simpler form than is typically presented in ZFC.
Jul
17
revised Theory of real numbers and using functions
added 12 characters in body
Jul
17
answered Theory of real numbers and using functions
Jul
14
comment What would be lost if ZF Axiom of Infinity replaced by Peano's Axioms?
@MaliceVidrine Hmmm... A good argument for an is-a-set predicate: Maybe not everything in set theory should be a set as in ZFC.
Jul
13
comment What would be lost if ZF Axiom of Infinity replaced by Peano's Axioms?
@MaliceVidrine What makes you think $x$ and $y$ they are both empty sets?
Jul
13
comment What would be lost if ZF Axiom of Infinity replaced by Peano's Axioms?
@MaliceVidrine No, you couldn't.
Jul
13
comment What would be lost if ZF Axiom of Infinity replaced by Peano's Axioms?
@MaliceVidrine You would never be able to prove things like $\exists x: x\in 1. $
Jul
13
comment What would be lost if ZF Axiom of Infinity replaced by Peano's Axioms?
@AndreasBlass I suspected as much. These "artifacts" were what motivated me to suggest the above alternative to AoI. I just wasn't sure how important the specific implementation of the axiom was. I thought, if all you want to do is postulate the existence of some infinite set, why not start with something really intuitive and useful like Peano's axioms, and avoid the above artifacts?
Jul
13
comment What would be lost if ZF Axiom of Infinity replaced by Peano's Axioms?
You wouldn't be able to prove stuff like $0\in 1$ or $2\subset 5$. Is that important?
Jul
13
comment What would be lost if ZF Axiom of Infinity replaced by Peano's Axioms?
So, could this terminology of yours (I am not familiar with it) be substituted for AoI without losing anything in ZFC theory?