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1h
revised What is the significance of using “$a$” vs “$x$” in this text?
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1h
answered What is the significance of using “$a$” vs “$x$” in this text?
1h
comment If every real valued continuous function on $X$ is uniformly continuous , then is every continuous function to any metric space uniformly continuous?
Nice example. $\;\!$
1h
comment If every real valued continuous function on $X$ is uniformly continuous , then is every continuous function to any metric space uniformly continuous?
@AlexRavsky, I think you mean: "put $A$ as the image of a Cauchy sequence that isn't convergent," but anyway I get your gist. I wonder if discrete implies uniformly discrete in a complete metric space.
1h
comment If every real valued continuous function on $X$ is uniformly continuous , then is every continuous function to any metric space uniformly continuous?
Does there exist a metric space $X$ and a subset of $A$ of $X$ such that $A$ is a discrete topological space under the induced topology, but $A$ is not a uniformly discrete subset of $X$?
14h
accepted Intuitionistically, are these inequivalent? $P \rightarrow Q,\; \neg Q \rightarrow \neg P,\; P \wedge \neg Q \rightarrow \bot,\; \neg P \vee Q$
21h
comment Name for the reals augmented with an $x$ such that $x^2 = x$
Another idea is to start with a join semilattice $L$, view $L$ as a commutative semigroup, and then take the $\mathbb{R}$-algebra freely generated by this commutative semigroup. Your algebraic structure is obtained in the special case where $L$ is the join semilattice $\{x\}$ defined by $x \vee x =x$.
21h
comment Name for the reals augmented with an $x$ such that $x^2 = x$
Strilanc, here's an idea for a generalization. We start with a poset $P$. We then take the (commutative) $\mathbb{R}$-algebra freely generated by the elements of $P$, and then quotient out by the relations $\{pq=q \mid p,q \in P, p \leq q\}.$ Your algebraic structure is obtained in the special case where $P$ is the poset with a single element $\{x\}$. Other interesting cases occur where $P$ is $\{x_0,x_1,x_2,\cdots\}$ totally-ordered in the obvious way.
23h
comment Name for the reals augmented with an $x$ such that $x^2 = x$
Doesn't this just show that we can embed the split complex numbers into the number system of interest? It doesn't show they're isomorphic.
1d
comment What is the name of this similarity measure for sets?
Interesting. Just out of curiosity, what is your motivation for considering that particular formula, as opposed to $|A \cap B|/\mbox{[something else]}$?
1d
answered How I can express in a pure symbolic way common reasoning? Examples inside.
1d
revised What is the future of Set Theory if it is NOT the foundation of Mathematics?
added 156 characters in body
1d
answered What is the future of Set Theory if it is NOT the foundation of Mathematics?
2d
asked Characterizing affine subspaces order-theoretically
2d
comment What do we call well-founded posets whose elements have a unique height?
Thanks, I think you might be right. Also, that paper looks relevant.
2d
asked What do we call well-founded posets whose elements have a unique height?
2d
answered Why Mendelson axiom schemas are true?
2d
comment Why Mendelson axiom schemas are true?
Better to say that $\varphi \implies \psi$ does not mean that "$\varphi$ causes $\psi$" or some such.
2d
comment Intuitionistically, are these inequivalent? $P \rightarrow Q,\; \neg Q \rightarrow \neg P,\; P \wedge \neg Q \rightarrow \bot,\; \neg P \vee Q$
@ZhenLin, so much for distinguishing between proof by contraposition and proof by contradiction in this way. How incredibly disappointing.
2d
comment Constructing the natural numbers without set theory.
@JohnColanduoni, how are functions in ETCS very different from those in type theory? They're both primitives.