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Apr
21
comment Non-constructive axiom of infinity
Hmmm ... with $A=\omega\cup\{\{42\}\}$ I get $\bigcup A=\omega\cup\{42\}\ne A$.
Apr
21
comment Non-constructive axiom of infinity
OK, I think I see now why your proof works, but I don't see where you use induction. As far as I can see you just build a contradiction by proving $\rank(\bigcup x)<\rank x$, which contradicts $\bigcup x=x$.
Apr
21
comment Non-constructive axiom of infinity
Ah, OK, that makes more sense. Now let me get back to understanding the rest. :-)
Apr
21
comment Non-constructive axiom of infinity
"We can prove that there are only finitely many sets of finite rank" ... but don't all natural numbers in the standard construction have finite rank? There are definitely infinitely many natural numbers (although each of them is of course a finite set in the standard construction).
Apr
21
comment n-dimension hypercube!
Duplicate? math.stackexchange.com/questions/763458/…
Apr
21
asked Non-constructive axiom of infinity
Apr
21
comment N-Dimension Hypercube question? (making sense of the question)
Are you sure you copied the complete text? Because it seems to me there's something missing (the grammar doesn't really work out well, either). I suspect the complete text contains something like "... to each of its vertex IDs and then we add new connections between pairs of vertices which differ only in the left-most bit."
Apr
21
comment Is it possible that “A counter-example exists but it cannot be found”
@TrevorWilson: Is the post now OK?
Apr
21
revised Is it possible that “A counter-example exists but it cannot be found”
Added a specific formalization to make the post correct
Apr
19
answered Visualizing the 48 actions of GL(2,3)
Apr
19
comment Is it possible that “A counter-example exists but it cannot be found”
@EricTowers: Actually the ordinal number version was exactly this interesting number paradox, with "natural" replaced by "ordinal" and "uninteresting" replaced by "undefinable". My point was that this paradox doesn't work with real numbers under the ordinary ordering, because other than non-empty sets of natural numbers, non-empty sets of real numbers (even bounded ones) don't need to have a minimum.
Apr
19
comment Is it possible that “A counter-example exists but it cannot be found”
@TrevorWilson: So the solution is to just demand a specific type of specification to be chosen, and then for this specific type of specification it is true? Or does the type of specification itself have to fulfil some requirements?
Apr
17
reviewed Approve suggested edit on how does this converges? Sequence and series convergence
Apr
17
comment Is there a more concise way?
@naslundx: Well, that must be the reason why Fermat could not write it on the border of the page. :-)
Apr
17
comment Is it possible that “A counter-example exists but it cannot be found”
Thinking again about my parenthetical comment in the previous comment: The well-ordering might itself be impossible to specify, therefore this trick probably doesn't work. Anyway, the fact that the very same argument can be applied to the ordinals and gives a contradiction already shows that the argument as such cannot be correct.
Apr
17
comment Is it possible that “A counter-example exists but it cannot be found”
Interesting. I can't say I understand all of what was discussed, but the example with the ordinals in the linked MathOverflow answer is quite convincing that something is wrong with the argument (@EricTowers: There exist nonempty sets of positive real numbers without a minimum — most notably the set of positive real numbers itself —, so this is no argument; however one could impose a well-ordering on the reals by mapping to an ordinal of the right cardinality and make the argument with that ordering). I'm not sure what I now should do with my answer. Just delete it? Amend it saying it's wrong?
Apr
17
awarded  Good Answer
Apr
16
awarded  Nice Answer
Apr
16
comment Find the solutions of: $\sin x+\cos x=\sin^2 x+0.5\sin{2x}$
The equation as written is wrong: Inserting $x=0$ gives $1=0$.
Apr
16
answered Is it possible that “A counter-example exists but it cannot be found”