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Results tagged with reverse-math
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user 4600
The general enterprise of calibrating the strength of classical mathematical theorems in terms of the axioms, typically of set existence, needed to prove them; originated in its modern form in the 1970s by H. Friedman and S. G. Simpson (see R.A. Shore, "Reverse Mathematics: The Playground of Logic", 2010).
25
votes
Accepted
What is the reverse mathematical strength of the fundamental theorem of algebra?
Tanaka and Yamazaki (in the volume Reverse Mathematics 2001, see review) show that a substantial portion of field theory can be done in the weak base theory RCA$_0$, by proving in RCA$_0$ the fundamen …
- 24.8k
20
votes
Accepted
Why is weak Kőnig's lemma weaker than Kőnig's lemma?
The issue is that for a finitely branching subtree $T$ of $\omega^{<\omega}$, the function $f$ mapping $\sigma$ to the greatest $n$ such that the concatenation $\sigma ^\frown n$ is in $T$ may not be …
- 24.8k
18
votes
Why is this new result such a big deal?
They show that $\DeclareMathOperator{\WKL}{WKL}\DeclareMathOperator{\RT}{RT}\DeclareMathOperator{\RCA}{RCA} \RT^2_2$ is $\Pi^0_3$-conservative over $\RCA_0$. Thus, there is no way that $\RT^2_2$ can b …
- 24.8k
10
votes
Proof complexity of two directions of equivalency?
This fits in the program of reverse mathematics. For instance: a subtree of the set of all finite binary strings has an infinite path iff it is infinite. One direction is provable in RCA $_0$ and the …
- 24.8k
8
votes
Accepted
Reverse Math of High Sets?
In the paper "On a conjecture of Dobrinen and Simpson regarding almost everywhere domination", Binns, Lerman, Solomon and I constructed $\omega$-models of this "high" principle which demonstrate it do …
- 24.8k
6
votes
Accepted
Is 0' of PA degree relative to a non-low set?
No, by the Arslanov completeness criterion $0'$ is only DNC (Diagonally non-computable) relative to low sets. And PA implies DNC.
- 24.8k
2
votes
Accepted
Proving boundedness of continuous images of [0,1] in WKL0
Here is a sketch of how the proof might go.
If $f:[0,1]\rightarrow\mathbb R$ is continuous but not bounded then the sets $S_n=[0,1]\setminus f^{-1}[(-n,n)]$ are closed with $S_n\ne\emptyset$ for all …
- 24.8k