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Jun 6, 2020 · Transfinite is good when there is a notion of order, so "transfinite ordinal", or when you want to talk about non-standard real numbers which are larger than all the standard natural numbers (in the context of non-standard analysis, that is), then "transfinite" is clearer than "infinite".
Jun 30, 2015 · Learning transfinite induction, and the theory of ordinals, is a bit time consuming. Many mathematicians are perfectly fine by learning this simple one lemma, and avoiding to learn all else. Someone correct me if I am wrong, but before Zorn's Lemma mathematicians based their arguments on the transfinite. Since then hardly anybody does this anymore.
Jun 30, 2015 · The principle of transfinite induction essentially says that, for a given formula P(x), if P(0) is true, and the truth of P(α) is preserved by taking successors and suprema, then P(α) must be true of all ordinals α. (We can omit the P(0) case because 0 = sup (∅).) Share. Cite. answered Jun 29, 2015 at 17:46.
Jun 13, 2020 · Or, when you construct by transfinite recursion a maximal chain in a partial order (using a choice function, of course), you need to prove by induction that the chain is maximal. And since the construction is a continuous construction (namely, taking union at limit steps), the work splits naturally to limit and non-limit.
Sep 21, 2016 · Transfinite induction is a proof technique. Transfinite recursion, on the other hand, is a construction technique. You use transfinite recursion to build some mathematical object (usually but not always a function), and you use transfinite induction to prove things about it. (Note that these terms often get conflated in the literature.)
Apr 9, 2022 · Theorem (Transfinite Construction). Let W be a well-ordered set, and E an arbitrary class. Assume: For each x ∈ W, there is a given rule Rx that associates with each φ: W(x) → E, a unique Rx(φ) ∈ E. Then there is one, and only one, F: W → E such that F(x) = Rx (F ∣ W(x)) for each x ∈ W.
You would want to use it whenever you are trying to establish that a property P P holds for all ordinals α α, in analogy with the ordinary induction case. The difference is that for ordinary induction, it only works up to the ordinal number ω ω and no further. Transfinite induction can be applied to higher ordinals.
11. The usage of "transfinite" there is not the same as what we now call "transfinite induction". Kolmogorov essentially just means "infinite". One example of the sort of thing that Kolmogorov calls "excluded middle" is now called the "limited principle of omniscience" (LPO). LPO says that if you have any property P(n) P (n) of a natural number ...
Our teacher gave us for practice to prove some properties of V(α) defined as V(0) = ∅, V(S(α)) = P(V(α)), Lim(α): V(α) = ⋃{V(β) | β <α} The properties are. 1) α <β ⇒ V(α) ⊆ V(β) 2) α ∈ V(S(α)) 3) α ∉ V(α) We should use transfinite induction to prove them. However, I've never seen a real application of transfinite ...
Oct 17, 2018 · Transfinite Recursion Theorem, Paramatric Version: Let be an operation. is a computation of length is a function such that and . Let be the property. Then defines an operation such that for all and for all sets z. Proof: defines an operation. We have to show that, for each , there is a unique such that . This is obvious for .
