Which [Python,] CAS support transfinite and/or surreal numbers and/or other piecewise axioms for an infinity symbol?
Are they vectorizable? What's wrong with substituting a standard symbol for infinity (instead of prematurely discarding e.g. coefficients/scalars and exponents)? https://github.com/sympy/sympy/wiki/SymPy-vs.-Sage#functionality https://github.com/sympy/sympy/wiki/SymPy-vs.-Sage#some-syntax-differences sympy.Symbol('Inf') sage. var('Inf') Should CPython try to be a CAS? On Mon, Oct 19, 2020, 7:36 AM Henk-Jaap Wagenaar <wagenaarhenkj...@gmail.com> wrote: > I have commented on Steven's comments about alephs below. > > It seems to me that this discussion (on having "different" infinities and > allowing/storing arithmetic on them) is dead-on-arrival because: > - the scope of people who would find this useful is very small > - it would change current behaviour > - it would be unusual/a first among (popular) programming languages* > - consistency is basically impossible: as somebody pointed out, if you > have a (Python) function that is 1 / (x ** 2), will the outcome be (1/0)**2 > or just 1/0? (1/0)**2 is the consistent outcome, but would require > implementing zeros with multiplicity... > > This would also create problems underlying, because Python floats (I > guess) correspond to the C and CPU floats, so performance and data > structure/storage (have to store additional data beyond a C(PU) float to > deal with infinities and their size, which would be unbounded if you e.g. > allowed exponentials as you would get into things like Cantor Normal Form ( > https://en.wikipedia.org/wiki/Ordinal_arithmetic#Cantor_normal_form)). > > I honestly think this way lies madness for the floats of a general purpose > programming language. > > * of course, somebody has got to be first! > > On Sun, 18 Oct 2020 at 23:03, Steven D'Aprano <st...@pearwood.info> wrote: > >> Oops, I messed up. (Thanks David for pointing that out.) >> >> On Sun, Oct 18, 2020 at 07:45:40PM +1100, Steven D'Aprano wrote: >> >> > Each of these number systems have related, but slightly different, >> > rules. For example, IEEE-754 has a single signed infinity and 2**INF is >> > exactly equal to INF. But in transfinite arithmetic, 2**INF is strictly >> > greater than INF (for every infinity): >> > >> > 2**aleph_0 < aleph_1 >> > 2**aleph_1 < aleph_2 >> > 2**aleph_2 < aleph_3 >> >> I conflated what I was thinking: >> >> # note the change in comparison >> 2**aleph_0 > aleph_0 >> 2**aleph_1 > aleph_1 >> 2**aleph_2 > aleph_2 >> ... > > >> which I think is correct regardless of your position on the Continuum >> Hypothesis (David, care to comment?), > > > Yes, due to Cantor's diagonal argument: > https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument > > > >> with this: >> >> 2**aleph_0 = aleph_1 >> 2**aleph_1 = aleph_2 >> 2**aleph_2 = aleph_3 >> ... >> >> which is only true if the Continuum Hypothesis is true > > > *generalized* Continuum Hypothesis, and in fact it isn't "only true if > (G)CH" is much stronger, it is in fact the same statement (if your "..." > means "for all ordinals" as GCH (CH is just the first one of those). See > https://en.wikipedia.org/wiki/Continuum_hypothesis#The_generalized_continuum_hypothesis > . > _______________________________________________ > Python-ideas mailing list -- python-ideas@python.org > To unsubscribe send an email to python-ideas-le...@python.org > https://mail.python.org/mailman3/lists/python-ideas.python.org/ > Message archived at > https://mail.python.org/archives/list/python-ideas@python.org/message/Z67R35LP7VT5UJA7F7ZTBGHTERK33IEJ/ > Code of Conduct: http://python.org/psf/codeofconduct/ >
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