The correspondence between assumptions and sets may not be quite straightforward to implement. In the original set theory of Cantor and Frege any condition (or statement) would define a set. After Russell remarked that "the set of all sets that do not belong to themselves" (if it exists) has contradictory properties, all later axiomatic set theories have restricted the correspondence between sets and statements in one way or another.
Kalevi Suominen On Monday, May 12, 2014 10:31:53 PM UTC+3, Aaron Meurer wrote: > > Another argument for keeping it is that it works nicely with the > assumptions system. In the assumptions system, ~Q.positive means "not > positive", whereas Q.nonpositive means "real and not positive". The > former item means *anything* that is not a positive real number. It > could be a negative number, a complex number, a matrix, a set, etc. In > other words, negation in the assumptions implicitly means universal > complement. Therefore if we want there to be a one-to-one mapping > between assumptions and sets (and I think we do), we need to allow the > universal set, because that is the corresponding set to Q.is_true (or > more simply, S.true). > > Aaron Meurer > > > On Sun, May 11, 2014 at 11:35 PM, Aaron Meurer <[email protected] > <javascript:>> wrote: > > I am -1 to remove it. As I explained in the issue, universal set is > > analogous to infinity for regular numerical arithmetic. Not all > > mathematical operations make sense for it, but there are some that do, > > and they are useful to have. Of course, sometimes you do want to > > specify your domain. It is similar with numbers: sometimes you want to > > specify explicitly that you are working with the reals or the > > rationals or whatever. But it is annoying to have to do this all the > > time. > > > > We should just fix code that assumes that the reals are the universal > set. > > > > Aaron Meurer > > > > On Sat, May 10, 2014 at 12:35 PM, Sergey B Kirpichev > > <[email protected] <javascript:>> wrote: > >> On Sat, May 10, 2014 at 11:03:12AM -0700, Harsh Gupta wrote: > >>> This comes from the discussions on this PR > >>> https://github.com/sympy/sympy/pull/7462#issuecomment-42111992 > >>> We have a class called UniversalSet which is supposed to be *set* > which > >>> contains all the sets which we can define in Sympy. The problem is > that we > >>> really don't "know" what our defined Universal Set is. It has been > >>> proposed by Sergey that we wipe out the Universal Set class. We can > >>> explicitly provide the the known defined universal set when > situation asks > >>> for it, that set can be complex, real or real*real or anything. I'm > +1 to > >>> the proposal. I guess this goes with "Explicit is better than > Implicit" > >>> from the Zen of Python. At places we are implicitly assuming > UniversalSet > >>> to Interval(-oo, oo) which is clearly wrong. Then we cannot define > >>> operations like PowerSet and cardinality on such sets. We can avoid > the > >>> problem by leaving PowerSet or cardinality undefined for > UniversalSet, but > >>> not having an UniversalSet will be a better way to avoid unknown > >>> inconsistencies. > >> > >> btw, some statistics: > >> $ fgrep -R UniversalSet sympy/|wc -l > >> 22 > >> $ fgrep -R UniversalSet sympy/ 2>/dev/null|grep -v 'core/sets.py'|grep > -v 'core/tests/test_sets.py'|wc -l > >> 8 > >> > >> -- > >> You received this message because you are subscribed to the Google > Groups "sympy" group. > >> To unsubscribe from this group and stop receiving emails from it, send > an email to [email protected] <javascript:>. > >> To post to this group, send email to [email protected] > <javascript:>. > >> Visit this group at http://groups.google.com/group/sympy. > >> To view this discussion on the web visit > https://groups.google.com/d/msgid/sympy/20140510183503.GB18695%40debian. > >> For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/5ea1ba36-4eb9-43d1-9cb7-8538a4635be5%40googlegroups.com. For more options, visit https://groups.google.com/d/optout.
