I was very struck by the following remark made in passing on Asaf Karagila’s blog:
We know absolutely nothing … I’m always pulled up short when reminded about gaps in our knowledge like this! Why are these things so hard?
I was very struck by the following remark made in passing on Asaf Karagila’s blog:
We know absolutely nothing … I’m always pulled up short when reminded about gaps in our knowledge like this! Why are these things so hard?
The way that remark was phrased raises a potentially interesting historical / sociological question: When / how was it decided that everyone would say “injection” and “surjection” instead of the far more intuitive “1-1” and “onto”?
I’m reminded of:
* The way everyone switched from the Wade-Giles transliteration of Chinese (Peking, Lao-tzu, Mao Tse-tung) to Pinyin (Beijing, Laozi, Mao Zedong).
* The Académie Française developing French words to replace English borrowings.
* An argument made by some Catholics that it’s better for the Nicene Creed to say “consubstantial with the Father” than the older Catholic and Book of Common Prayer’s “of one substance with the Father” because it’s a divine mystery (μυστήριον) and “consubstantial” makes it harder for people to think they understand it (a sort of defamiliarisation effect).
(Before Vatican II, the Catholic translation was “of one substance with the Father”; then it became “one in being with the Father”. Finally, in the new translation developed under Benedict XVI it became “consubstantial with the Father”.)
Blame Bourbaki? (Injective/surjective is their usage, see Theory of Sets p. 84.)
I didn’t write it there, but we know that the Partition Principle implies the Axiom of Choice from families which can be well-ordered (which is equivalent to the statement “if there is a surjection onto an ordinal, then there is an injection from the ordinal as well”, i.e. PP where B is required to be well-ordered).
We also know that the aforementioned choice principle implies Dependent Choice.
And we know that the Weak Partition Principle implies that there are no infinite Dedekind-finite sets.
The reason, I suspect, these are so hard to work out, is that the implications are in fact strict. But we don’t have any tools to construct models where they fail. Yet. We don’t have these yet.