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Apr
6
comment How do I prove cardinality is well-defined?
@DanielEscudero Another thing I forgot to mention about your response is that it should probably be checked that $I_n\not\sim I_m$ if $n\neq m$. That is implicit in what your method needs for well-definedess (but not what was needed in user45150's original question)
Apr
6
comment How do I prove cardinality is well-defined?
@user42912 I think what is important to remember with regards to Daniel Escudero's definition is that it is a way to associate a size, called the cardinality, to every set $X$. For this you must check that $\sim$ is an equivalence relation because if you associate the number $n$ to $X$ and $m$ to $Y$ and $X\sim Y$ you would like to be able to conclude that $n=m$.
Apr
6
comment How do I prove cardinality is well-defined?
@DanielEscudero The discussion you give above may be a little confusing, because you do not mention what $\text{card}(X)$ actually is. For instance what happens if $X$ is infinite. I agree that the conflation of the definition of cardinality that you give and the one in the original question may be the cause of the confusion. The reason I did not mention this point of view in my comments was that it was not mentioned in the question and I did not want to add confusion, but since it is brought up we may as well develop it.
Apr
4
comment Understanding algebraic closure
On the $[K:F_p]$ point. Notice that $K$ must be countably infinite, and since $F_p$ is finite $[K:F_p]$ must be the cardinality of the natural numbers
Apr
4
comment How do I prove cardinality is well-defined?
The notion is well-defined already, even before you prove it is an equivalence relation. There is nothing to check. This is because the definition: "there exists a bijection" doesn't depend any choice or anything, either there is one or there isn't
Apr
4
comment How do I prove cardinality is well-defined?
I guess I am confused. $\text{card}(X)=\text{card}(Y)$ if and only if there exists a bijection between $X$ and $Y$. This is a legitimate definition. You don't pick a representative for an equivalence set or any other of the usual well-definedness problems.
Apr
4
comment How do I prove cardinality is well-defined?
What do you mean by welldefined? Do wish to show that it is an equivalence relation?
Mar
26
comment Ring homomorphisms on the set of rationals that coincide on integers
Taking $1$ to $1$ is in the definition of ring homomorphism. See for instance:en.wikipedia.org/wiki/Ring_homomorphism, it seems to be standard. Perhaps you are trying to use another definition and that is the confusion?
Mar
25
comment Ring homomorphisms on the set of rationals that coincide on integers
Ring homomorphism by definition takes $1$ to $1$
Mar
24
comment Ring homomorphisms on the set of rationals that coincide on integers
If $f$ is not the zero map, the image of $f$ is a copy of $\mathbb{Q}$. Note that $f(n)=nf(1)=n\cdot 1=n$. How can you use that the image of $f$ is isomorphic to $\mathbb{Q}$ and the fact that $n$ is in the image?
Mar
23
comment Ring homomorphisms on the set of rationals that coincide on integers
Then use Josh Keneda's comment to allow you to divide by $n$.
Mar
21
comment Ring homomorphisms on the set of rationals that coincide on integers
To add to my comment. What happens if you multiply $f(1/n)$ and $g(1/n)$ by $n$?
Mar
21
comment Ring homomorphisms on the set of rationals that coincide on integers
Notice that if you add $p/q$ to itself $q$ times you get $p$. How does this help?
Mar
20
comment Is this a theorem or a conjecture?
I don't think that is the claim. The claim being made is more subtle. The claim you state is basically the density of the rationals. Also $[1/3,1/2]$ is definitely not in the form you state
Mar
19
comment Can we avoid an axiomatic theory of sets by never formulating paradoxes?
I guess to clarify even more, I mean that natural numbers shouldn't really have precedence of ontology over other finite objects, like formulae if we are thinking about natural numbers as strokes.
Mar
19
comment Can we avoid an axiomatic theory of sets by never formulating paradoxes?
Sure. My point was more on one that natural numbers don't really have a precedence of ontology
Mar
19
comment Can we avoid an axiomatic theory of sets by never formulating paradoxes?
Asaf, for 2 surely if we can write down strokes and call them natural numbers, we can write down sentences and call them formulae without any need for Godel encodings?
Mar
19
comment Can we avoid an axiomatic theory of sets by never formulating paradoxes?
I think Asaf's first paragraph is a good analogy for what I was trying to say
Mar
19
comment Can we avoid an axiomatic theory of sets by never formulating paradoxes?
The paradoxes show that a statement and its negation can be proved in that system. The fear is that since you know you can generate a paradox, that you may unknowingly use one in your proof (though maybe not one of the known ones) and therefore may be able to prove a statment $\phi$ and its negation $\lnot \phi$. ZFC is a formalization of what rules you can use and seemingly using those rules does not lead to a paradox (though this has not be proved and can't really be proved, but that is another discussion)
Mar
17
comment Is $\prod \limits_{i = 1}^{n} [0,1] \subseteq \mathbb R^n$ homeomorphic to the closed unit ball?
There is probably still things to check. For instance, continuity is clear everywhere but at zero, where you have to use that $d_x$ is bounded.