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Aug
19
comment Indirect proof , odd and even numbers
Are you familiar with the idea of "contrapositive"? I.e. showing "if A then B" by showing "if not B, then not A"?
Aug
19
comment Why does $(a+b)^2= a^2+b^2 + 2ab$? Why is the $2ab$ there?
Are you sure it said $f(x + h)^2 - f(x)^2$, as opposed to $f(x + h) - f(x)$?
Aug
18
comment Does the Cartesian product of an infinite family have all the elements we expect?
Andres, I'm looking for an answer like "yes, that products are nonempty implies that any element satisfying certain conditions appears in the set", or "yes, the size of the resulting set is at least $X$", or "no, you generally can't say anything stronger than the existence of a single element".
Aug
18
comment Does the Cartesian product of an infinite family have all the elements we expect?
Asaf, is it correct to say that the axiom of choice lets you use the fact that "such a set exists" (in addition to the more commonly cited fact that "such a set is non-empty")?
Aug
18
asked Does the Cartesian product of an infinite family have all the elements we expect?
Aug
17
comment Equivalence between “mathematical induction” and “transfinite induction” for natural numbers?
There was definitely induction involved in the proof that $\mathbb{N}$ is well-ordered--I'm not sure how to resolve this issue either. However, I'm okay with leaving the issue alone, and I'm happy with your answer--my main goal here was to get help with formulating the equivalence sensibly.
Aug
17
accepted Equivalence between “mathematical induction” and “transfinite induction” for natural numbers?
Aug
15
comment Equivalence between “mathematical induction” and “transfinite induction” for natural numbers?
Thanks Thomas, those were oversights on my part when I typed the question.
Aug
15
revised Equivalence between “mathematical induction” and “transfinite induction” for natural numbers?
added 30 characters in body
Aug
15
comment Equivalence between “mathematical induction” and “transfinite induction” for natural numbers?
Thanks, fixed my mistake.
Aug
15
asked Equivalence between “mathematical induction” and “transfinite induction” for natural numbers?
Aug
13
comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number?
Yup, this suffices. Let $S$ be the set of natural numbers for which every transitive subset is a natural number. $0 \in S$ since the only possible subset of $0$ is just $0$. Suppose $n \in S$, and let $x$ be a transitive subset of $n^+$. If $n$ is not an element of $x$, then $x \subset n$, and we just use the inductive hypothesis. Otherwise, $n \in x$, and transitivity implies $n \subset x$, so in fact $n^+ \subset x$, and $n^+ = x$. Either way, $x$ is a natural number, and the proof is complete.
Aug
13
comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number?
That said, all this discussion has been immensely helpful anyway! I ended up doing something like this: (1) prove by induction that if X is a non-empty subset of a natural number, then the intersection of X is a natural number; (2) find a set F such that the intersection of F is the same as the intersection of E, but F is a subset of a natural number
Aug
13
comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number?
Martin, I'm still not fully convinced, since to me it looks like you're using the assumption that $a \in n$ if and only if $a$ is a proper subset of $n$ before you've proved that $a$ is a natural number.
Aug
13
comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number?
Asaf, does math.stackexchange.com have a prediction market for estimates like this (e.g. one where you can wager reputation points)? :)
Aug
13
comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number?
Martin, this is indeed how the natural numbers are defined in the text I'm referencing (Halmos - Naive Set Theory).
Aug
13
comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number?
I feel like this should suffice, but I still need to convince myself that "a transitive set of natural numbers which is a subset of some natural number is itself a natural number".
Aug
13
comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number?
Thanks Martin! I agree that $a \subset n$ for each $n \in E$, but how did you then conclude that $a \in n$ for each $n \in E$?
Aug
13
comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number?
Unfortunately I need to use this to prove that E has a smallest entry in the first place, so I can't just say "take the smallest entry of E".
Aug
13
asked Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number?