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| bio | website | |
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| location | ||
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| visits | member for | 5 months |
| seen | Feb 27 at 10:12 | |
| stats | profile views | 15 |
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Dec 20 |
answered | Is there a bounded-quantifier formula in ZF without Foundation that defines the naturals? |
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Dec 20 |
comment |
Is there a bounded-quantifier formula in ZF without Foundation that defines the naturals? @Asaf Karagila. I am a bit confused. Let us take the axiom foundation: $$\forall z [z\neq \emptyset \rightarrow (\exists y\in z) y\cap z=\emptyset]$$ The formula that says "$x$ is well-founded by $\in$" is: $$(\forall z\subseteq x) [z\neq \emptyset \rightarrow (\exists y\in z) y\cap z=\emptyset]$$ How do you eliminate the $(\forall z\subseteq x)$ at the beginning and get a bounded formula? |
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Dec 20 |
comment |
Is there a bounded-quantifier formula in ZF without Foundation that defines the naturals? @Asaf Karagila. How do you write "$x$ is well-ordered by $\in$" as a bounded formula over ZF-Foundation? |
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Dec 13 |
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If $\alpha$ is an indecomposable ordinal, why $\Gamma(\alpha\times{\alpha})=\alpha$? @Camilo Arosemena. You're welcome. Can you please post any errors you find. |
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Dec 13 |
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An exercise from Levy's Basic Set Theory (Exercise 3.17) Thank you for your help. Unfortunately, I fail to see why $\varphi$ is order-preserving. Let $\sigma = (5,5,5,5)$, $\tau = (1,1)$, $\xi = 1$, $\eta = 3$. Now $\xi < \eta$ but $\sigma^\frown \xi <_l \tau^\frown \eta$ fails to hold. Indeed, $$\tau^\frown \eta = (1,1,3) <_l (5,5,5,5,1) = \sigma^\frown \xi.$$ |
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Dec 11 |
revised |
If $\alpha$ is an indecomposable ordinal, why $\Gamma(\alpha\times{\alpha})=\alpha$? corrected grammar |
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Dec 11 |
awarded | Supporter |
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Dec 11 |
revised |
If $\alpha$ is an indecomposable ordinal, why $\Gamma(\alpha\times{\alpha})=\alpha$? corrected spelling |
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Dec 11 |
revised |
If $\alpha$ is an indecomposable ordinal, why $\Gamma(\alpha\times{\alpha})=\alpha$? corrected spelling |
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Dec 11 |
answered | If $\alpha$ is an indecomposable ordinal, why $\Gamma(\alpha\times{\alpha})=\alpha$? |
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Dec 7 |
awarded | Teacher |
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Dec 7 |
answered | Textbooks on set theory |
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Dec 5 |
comment |
An exercise from Levy's Basic Set Theory (Exercise 3.17) I added his statement to my original post. |
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Dec 5 |
comment |
An exercise from Levy's Basic Set Theory (Exercise 3.17) Mathematicians use paradox for surprising and unexpected results (for example Banach–Tarski paradox, Skolem's paradox). Milner-Rado paradox fails for finite unions. I guess that is why the infinite case is unexpected. |
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Dec 5 |
revised |
An exercise from Levy's Basic Set Theory (Exercise 3.17) added the exercise and the hint from Levy's book. |
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Dec 5 |
comment |
An exercise from Levy's Basic Set Theory (Exercise 3.17) Thank you very much for the clarification about the order. Unfortunately, I am still unable to complete the proof. But still working on it. |
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Nov 28 |
awarded | Analytical |
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Nov 28 |
awarded | Editor |
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Nov 28 |
comment |
An exercise from Levy's Basic Set Theory (Exercise 3.17) Fixed it. Thanks. |
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Nov 28 |
revised |
An exercise from Levy's Basic Set Theory (Exercise 3.17) changed 0004 to 00004 |
