11
votes
Accepted
Is it OK if a proposition contains $\frac{1}{0}$?
As with many questions in logic, the answer to this question is "it depends on how you set things up".
Most questions in math don't depend on the precise details of your foundational system, ...
- 35.6k
5
votes
Accepted
Does $\omega$-consistency depend on the encoding?
This is a great question! In general, "reasonably strong" theories will always have distinguished implementations (the jargon is actually interpretations, but meh) of the natural numbers, ...
- 237k
3
votes
Why do Dedekind cuts use sets?
"Ranges" are just sets; the "range" of all numbers less than $1$ is (in $\Bbb R$) the set $(-\infty,1)$. On these grounds, I fail to see the issue; I somewhat agree that Dedekind ...
- 32.6k
3
votes
How should one understand the "universe of sets"?
You may be interested in reading some of Nik Weaver's stuff, maybe Is set theory indispensable? and The concept of a set for starters. If I interpret his position correctly I think he agrees with you ...
- 409k
2
votes
Accepted
Is there reason for concern with this proof that "there is a bijective function from $\{1,...,m\}$ to $\{1,...,n\}$ only if $m = n$"?
In the context of proving a statement like this, you are correct that you should really never be using the notation "Card" as if it were a function. But the proof still works when you ...
- 323k
1
vote
Is it OK if a proposition contains $\frac{1}{0}$?
Let's first ask, what is a fraction from the perspective of classical algebra and ring theory.
We pick a ring $R$ (a set which has addition, substraction, multiplication; think about integers, $\...
- 2,211
1
vote
Is it OK if a proposition contains $\frac{1}{0}$?
Let's simplify the problem. Suppose we have at least $0\in X$ and $1\in X$, and $f$ such that:
$~~~~~~\forall a:[a\in X \land a\neq 0 \implies f(a)=1]$
This definition of function $f$ does not give us ...
- 14.1k
1
vote
Is it OK if a proposition contains $\frac{1}{0}$?
If you wish to simultaneously be extremely rigorous and practical, a way out is simply define
$$\frac x 0 = 0$$
because division is just another operation and we can define it as whatever we want, as ...
- 3,985
1
vote
Is it OK if a proposition contains $\frac{1}{0}$?
is something like $$0\neq 0 \Rightarrow \frac{)+>1/}{=\in \subset}\mathbb{Q}\tag1$$ also a proposition?
The string of symbols $(1)$ is well formed according to first-order logic's syntactic rules, ...
- 38k
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