Sujet : Re: Replacement of Cardinality
De : invalid (at) *nospam* example.invalid (Moebius)
Groupes : sci.logic sci.mathDate : 15. Aug 2024, 22:25:23
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <v9lro3$13gee$1@dont-email.me>
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Am 15.08.2024 um 21:37 schrieb Moebius:
Am 15.08.2024 um 21:28 schrieb Jim Burns:
Defining ⅟𝔊 to be the smallest unit fraction
isn't a claim that ⅟𝔊 exists.
You need an existence proof [...] BEFORE stating a/the proper definition.
When defining a term we have to observe certain "definition rules".
For example, if you want to define/introduce the constant c as
c is the xxx
you have 1. to prove that there IS an xxx (->existence) and 2. that there is not more than one xxx (->uniqueness).
Otherwise one risks to introduce a contradiction by the definition. (This NOT something we would like to do in math.)
'⅟𝔊' can be used to derive contradictions,
Well, ... not really. See above.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For example, we can't define/introduce "c" the following way:
c = the real number x such that x * 0 = 1.
(or c = x iff x e IR and x * 0 = 1).
You see, after defining c that way, we could derive
c e IR & c * 0 = 1.
From this we could derive the "theorem":
Ex e IR: x * 0 = 1.
Which clearly is not the case.
Please try to understand that a definition is not an "assumption". Hence if we get a contradiction (after using a "definition" which is not proper) we can't just "discarge" that definition (as if it were an assumption).