The only thing missing here is a little anecdote about how Von Neumann, when challenged on this, did it in his head and started rattling off the parameters. For arbitrary animals.
Source please. Must learn more.
How to fit an elephant (2011)
11–20 of 27 posts
Re: How to fit an elephant (2011)
#12I can't help but feel like a complex number is two parameters (real&imag / mod&arg) - so really this is 8 parameters.
Cantor showed that these sets have the same cardinality. You can represent a complex (in the form of two reals) by interleaving digits or using a space-filling curve for example.
>
Cantor showed that these sets have the same cardinality. You can represent a complex (in the form of two reals) by interleaving digits or using a space-filling curve for example.
But this (set) isomorphism between R and C is not continuous. Indeed one can show that there exists no continuous epimorphism f: R^n -> R^m, where m > n, since for every such continuous map f the image f(R^n) has a measure of 0 with respect to the Borel measure in R^m.
Re: How to fit an elephant (2011)
#13I can't help but feel like a complex number is two parameters (real&imag / mod&arg) - so really this is 8 parameters.
He could have used an octonion, then it was 1 parameter.
Re: How to fit an elephant (2011)
#14The only thing missing here is a little anecdote about how Von Neumann, when challenged on this, did it in his head and started rattling off the parameters. For arbitrary animals.
Source please. Must learn more.
Von Neumann was renowned for his great prowess at mental maths. A famous (if also not entirely serious) story:
"When posed with a variant of this question involving a fly and two bicycles, John von Neumann is reputed to have immediately answered with the correct result. When subsequently asked if he had heard the short-cut solution, he answered no, that his immediate answer had been a result of explicitly summing the series (MacRae 1992, p. 10; Borwein and Bailey 2003, p. 42)."
Re: How to fit an elephant (2011)
#15Earlier quoted context omitted.
Cantor showed that these sets have the same cardinality. You can represent a complex (in the form of two reals) by interleaving digits or using a space-filling curve for example.
> Cantor showed that these sets have the same cardinality. You can represent a complex (in the form of two reals) by interleaving digits or using a space-filling curve for example. But this (set) isomorphism between R and C is not continuous. Indeed one can show that there exists no continuous epimorphism f: R^n -> R^m, where m > n, since for every such continuous map f the image f(R^n) has a measure of 0 with respec…
> there exists no continuous epimorphism f: R^n -> R^m, where m > n
Really? Then what is a https://en.wikipedia.org/wiki/Space-filling_curve?
Re: How to fit an elephant (2011)
#16Earlier quoted context omitted.
> Cantor showed that these sets have the same cardinality. You can represent a complex (in the form of two reals) by interleaving digits or using a space-filling curve for example. But this (set) isomorphism between R and C is not continuous. Indeed one can show that there exists no continuous epimorphism f: R^n -> R^m, where m > n, since for every such continuous map f the image f(R^n) has a measure of 0 with respec…
> there exists no continuous epimorphism f: R^n -> R^m, where m > n Really? Then what is a https://en.wikipedia.org/wiki/Space-filling_curve ?
Right from that article you linked: "A non-self-intersecting continuous curve cannot fill the unit square because that will make the curve a homeomorphism from the unit interval onto the unit square (any continuous bijection from a compact space onto a Hausdorff space is a homeomorphism). But a unit square has no cut-point, and so cannot be homeomorphic to the unit interval, in which all points except the endpoints are cut-points."
Re: How to fit an elephant (2011)
#17Earlier quoted context omitted.
> there exists no continuous epimorphism f: R^n -> R^m, where m > n Really? Then what is a https://en.wikipedia.org/wiki/Space-filling_curve ?
Right from that article you linked: "A non-self-intersecting continuous curve cannot fill the unit square because that will make the curve a homeomorphism from the unit interval onto the unit square (any continuous bijection from a compact space onto a Hausdorff space is a homeomorphism). But a unit square has no cut-point, and so cannot be homeomorphic to the unit interval, in which all points except the endpoints a…
Yeah, a non-self-intersecting map cannot, but OP didn't specify that, only "epimorphic" which certainly can.
Moreover OP's argument specifically proves too much, because space-filling curves (as described in the article) have a range with positive Borel measure.
Re: How to fit an elephant (2011)
#18I can't help but feel like a complex number is two parameters (real&imag / mod&arg) - so really this is 8 parameters.
Cantor showed that these sets have the same cardinality. You can represent a complex (in the form of two reals) by interleaving digits or using a space-filling curve for example.
Pairing functions only work on countable sets. This is funny because Cantor is the same person who proved real numbers are uncountable, and that there is no pairing function between 1 real number and naturals, let alone 2.
Re: How to fit an elephant (2011)
#19Earlier quoted context omitted.
Cantor showed that these sets have the same cardinality. You can represent a complex (in the form of two reals) by interleaving digits or using a space-filling curve for example.
Pairing functions only work on countable sets. This is funny because Cantor is the same person who proved real numbers are uncountable, and that there is no pairing function between 1 real number and naturals, let alone 2. https://en.m.wikipedia.org/wiki/Countable_set
I'm claiming a "pairing function" between single reals and pairs of reals. They have respective cardinalities 2^N0 and 2*2^N0=2^N0 where N0<2^N0 is the cardinality of the naturals.
Re: How to fit an elephant (2011)
#20Earlier quoted context omitted.
Cantor showed that these sets have the same cardinality. You can represent a complex (in the form of two reals) by interleaving digits or using a space-filling curve for example.
> Cantor showed that these sets have the same cardinality. You can represent a complex (in the form of two reals) by interleaving digits or using a space-filling curve for example. But this (set) isomorphism between R and C is not continuous. Indeed one can show that there exists no continuous epimorphism f: R^n -> R^m, where m > n, since for every such continuous map f the image f(R^n) has a measure of 0 with respec…
Sure, but did we need continuity? Also, if you want to be awkward, you can get around this by using the discrete topology, I don't think we needed the metric structure of R^n.