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Jul
2
comment Cartesian product with all elements
The order does matter. I find it useful to think of the Cartesian product of sets $A$ and $B$ as follows: $\forall x,y:[(x,y)\in A\times B \iff x\in A\land y\in B]$.
Jul
1
comment If $P$ is true then not $Q$
It seems to me that "This sentence is true" is meaningless nonsense. What then are we to make of the statement $P\implies Q$ where $P$ is meaningless nonsense?
Jun
30
comment Principle of Transfinite Induction
Having trouble googling this topic. Can you give me an online reference to a proof?
Jun
30
comment Principle of Transfinite Induction
Can the supremum (least upper bound) of the natural numbers be constructed in ZFC?
Jun
30
comment Implies in a truth table, unclear.
Just remember that, in mathematics, if not in everyday usage, $P\implies Q$ just means that $\neg[P\land \neg Q]$.
Jun
29
revised Rigorous proof of '${\{A \Rightarrow B}\} \iff {\{\neg B \Rightarrow \neg A}\}$' for a high school student
added 163 characters in body
Jun
29
answered Rigorous proof of '${\{A \Rightarrow B}\} \iff {\{\neg B \Rightarrow \neg A}\}$' for a high school student
Jun
29
revised Arithmetic functions
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Jun
29
revised Arithmetic functions
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Jun
29
revised Arithmetic functions
added 140 characters in body
Jun
29
revised Arithmetic functions
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Jun
28
comment The recursion principle.
Not familiar with your notation. Assuming you want to prove the existence of unique $a:N\to N$ such that: $a(0)=c$ and $a(n+1)=f(n,a(n))$. I would construct the set $a$ of ordered pairs of natural numbers such that: $\forall m,n:[(m,n)\in a \iff (m,n)\in N^2 $ $\land \forall d\subset N^2:[(c,0)\in d \land \forall p,q:[(p,q)\in d \implies (p+1,f(p,q))\in d]] \implies (m,n)\in d]$ Then prove $a$ is the required function.
Jun
27
revised Universal introduction - how exactly does it work?
edited body; edited title
Jun
24
comment Are proofs by induction inferior to other proofs?
@CameronWilliams Did you miss my last sentence?
Jun
24
answered Are proofs by induction inferior to other proofs?
Jun
23
revised Predicate logic by resolution
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Jun
23
revised Predicate logic by resolution
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Jun
23
answered Predicate logic by resolution
Jun
22
revised Does mathematics become circular at the bottom? What is at the bottom of mathematics?
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Jun
22
revised Does mathematics become circular at the bottom? What is at the bottom of mathematics?
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