Elliott
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 Aug17 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. Aug17 accepted Equivalence between “mathematical induction” and “transfinite induction” for natural numbers? Aug15 comment Equivalence between “mathematical induction” and “transfinite induction” for natural numbers? Thanks Thomas, those were oversights on my part when I typed the question. Aug15 revised Equivalence between “mathematical induction” and “transfinite induction” for natural numbers? added 30 characters in body Aug15 comment Equivalence between “mathematical induction” and “transfinite induction” for natural numbers? Thanks, fixed my mistake. Aug15 asked Equivalence between “mathematical induction” and “transfinite induction” for natural numbers? Aug13 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. Aug13 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 Aug13 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. Aug13 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)? :) Aug13 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). Aug13 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". Aug13 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 $n \in E$? Aug13 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". Aug13 asked Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number? Aug13 comment Intersection of a non-empty set of natural numbers (set-theoretic definition) gives an element of that set? I agree that "every element of a natural number is a natural number", but how does it follow that the intersection of a set of natural numbers is a natural number? Aug13 comment Intersection of a non-empty set of natural numbers (set-theoretic definition) gives an element of that set? Thanks jfhc! This makes sense. Are we required to consider $m^+$ though? Could we instead have said "if m is a proper subset of a for every a in E, then m is an element of a for every a in E, so m is fact an element of the intersection, which is not allowed; thus m must be equal to a for some a in E"? Aug13 comment Intersection of a non-empty set of natural numbers (set-theoretic definition) gives an element of that set? Wait a second, E is not a set of ordinals, E is a set of natural numbers! Aug13 comment Intersection of a non-empty set of natural numbers (set-theoretic definition) gives an element of that set? Yes, we've proved this already. Aug13 asked Intersection of a non-empty set of natural numbers (set-theoretic definition) gives an element of that set?