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Sep
1
answered How can you prove the equivalance relation for the following model?
Aug
30
comment Is $\aleph_0 = \mathbb{N}$?
If it's not first order then it's not generally called "Peano arithmetic", it's just "arithmetic".
Aug
30
comment Is $\aleph_0 = \mathbb{N}$?
Not to mention I think you missed the point. Curiosity was expressed about implementations of $\aleph_0$ that differ from one for $\mathbb{N}$; I was offering an example of the converse, which is easier to find, not recommending the Zermelo naturals.
Aug
30
comment Is $\aleph_0 = \mathbb{N}$?
Peano's axioms by themselves? That wouldn't be very helpful if analysis is your concern.
Aug
30
revised tautologies and truth values
complex-analysis tag is not appropriate for the question.
Aug
30
comment Is $\aleph_0 = \mathbb{N}$?
Or work in $\mathsf{NFU}$ where urelements have a mathematical purpose and you can use the intuitively appealing Frege natural numbers. With some other trade-offs along the way... :P
Aug
30
comment Is $\aleph_0 = \mathbb{N}$?
Is it really that bizarre? Does it ever even come up? Is it any weirder than reals as Dedekind cuts?
Aug
30
comment Is $\aleph_0 = \mathbb{N}$?
Let $\mathbb{N}'$ be $\bigcap\{x:\emptyset\in x\wedge\{\{z\}:z\in x\}\subseteq x\}$. One can recursively define all the usual arithmetic, and its not identical with $\aleph_0$. But that's not the set that's conventionally used for the naturals.
Aug
30
comment Is $\aleph_0 = \mathbb{N}$?
@goblin: I would argue that in naive mathematics the question simply isn't asked, not that is answered in the negative. This is certainly the case in type theories (not being able to phase it in good syntax is not a negative answer).
Aug
30
comment Is $\aleph_0 = \mathbb{N}$?
@GitGud: There may be only one decent version of cardinality in ZFC, but there is at least one other implementation of $\mathbb{N}$ that's not totally awful.
Aug
27
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
@Dan: You only need the one. If there are any other inductive sets, their intersection with X is a subset of X.
Aug
27
comment how to define a function?
For clarification, if you're defining a function without set theory, what are you defining it in? And what is the "it" in "But if we define a function without set theory it states that..."?
Aug
20
comment The empty set is a neighborhood?
Those examples are from the "open sets definition" section of the page. The sets in those examples are the open sets of a topology, not its neighborhoods.
Aug
16
comment Set Theory ZF Axioms Doubt
@HenningMakholm: In Russellian type theory you might more accurately say "each number one is a set," since there are infinitely many instances of $\mathbb{N}$, for which claims of isomorphism wouldn't even be formulae of the language.
Aug
11
answered What relations compose the language of ZFC?
Aug
8
comment What is a Primitive Atomic Formula?
Atomic formulae consist of a basic predicate symbol of the language and the appropriate number of terms. It has no bearing on the natural language interpretability of the predicate.
Aug
5
comment Is there a purpose behind a function?
Same reason we have names for specific mammals instead of just calling dogs, cats, people, and wombats "mammal".
Jul
26
comment Is there a rule for uniform substitution of predicate symbols in FOL?
Substitution of predicates as you ask about is guaranteed to work by the fact that the logical axioms (I'm going to assume an axiomatic presentation) are defined to be substitution instances of certain schemata (and it is completely unrelated, as you guess, to quantification over predicates). I think what you're wondering is whether there's a specific rule of inference that lets you use that sort of substitution as a line of a proof, and there's generally not, for the reason that you can always write down a logical truth (wrt whatever logical system you're using) as a line in your proof.
Jul
26
comment Set-builder Notation
I don't know what the purpose of 1 and 2 are, but generally "$y\in \{x:\phi\}$" is just notation for "$\exists z(y\in z\wedge \forall x(x\in z \leftrightarrow \phi))$", so 3 is an immediate consequence.
Jul
18
comment Order in writing composed morphisms
For what it's worth, Paul Taylor uses $f;g$ as a synonym for $g\circ f$ in Practical Foundations of Mathematics. I thought it was a nice convention to allow the more understandable ordering without it getting confused with the traditional ordering.