Zero equals infinity may be hard to swallow. However, non-self-identical (i.e., non-selfsame) numbers do not even require infinite sets; infinite sequences are sufficient. What, for instance, is the limiting sum of the following divergent series?
¥ - ¥ + ¥ - ¥ + ¥ - ¥ + ¥ -…
It could be read as:(¥ - ¥) + (¥ - ¥) + (¥ - ¥) + …
Which is simply 0 + 0 + 0 = 0.
Or it could read as:
¥ - {(¥ - ¥) + (¥ - ¥) + (¥ - ¥) + …}
Which is simply ¥ - {0 + 0 + 0 + …} = ¥ .
Therefore, 0 = ¥ .
In collective mind of the mathematical establishment, such non-self-identical numbers (i.e., zero-identical-to-infinity-identical-to-zero) were consensually banished in the late 19th century by insistence that limit sums of divergent series cannot properly be specified without simultaneously stating the procedure by which such limits are to be calculated. Self-consistency, which is an essential attribute of selfsameness, is a tacit assumption and deep emotional commitment of those persons who do not experience animistic identity transparency. Wherever inconsistency in mathematics has been encountered by such persons, more rules have been generated, old rules have been changed, new levels of consideration have been postulated so as to resurrect the lost consistency - because, without consistency, the integrity of egoic, individualistic, selfsame, selfhood cannot be psychologically sustained. And without such selfhood, its psychological projection, the nation-state, cannot be legitimized. And without the discipline of the state, we would all be back in agonies of the tribe, in horrors of “each against all”: fear of the non-selfsame is fear of the collective unconscious, the Dark Continent, source of all psychological projections. One, of course, a selfsame one, that is, does not recognize self-organized criticality of the tribe, mediated by relative-state of the corpus: participation mystique.
Instances of non-selfsameness were also dissipated by banishing the threat of multivalent logics with their manifest (from the point of view of single-valued logic) inconsistencies. Tarskian logical semantics (which defines a hierarchy of metalevels distinguishing “statements in” from “statements out and about”, i.e., statements “in” mathematics from statements “about” mathematics: nested metalanguages) was created specifically to banish m-valued logics and the non-self-identical numbers such logics give rise to. Why banish? Because non-selfsame numbers are the ultimate inconsistency nightmare for persons wed to worldview constructs based upon the notion that Nature is identical to itself, selfsame, and that, therefore, there can be only one correct account of it -- that notion, that is, most responsible for conflict between cultures throughout human history. Each culture believes it has been vouchsafed the only true account of selfsame reality, and, therefore, by matter of the fundamental circumstances of life finds itself in conflict with all other cultures. “Pure” mathematics has resolutely devoted itself to fostering and sustaining this source of warfare.