Correction: (max 500 charact.)
The complaint will not be published.
I 69
Intuitionism/Waismann: allows only proofs that can be constructed in a finite number of steps (that are constructive). All others are meaningless.
Formalism/Waismann: also allows non-constructive proofs. This dispute is, however, idle, if it is true that the word existence has no clear meaning from the outset. It is only obtained by the proof. And then a corresponding different one.
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Intuitionism .
I 74
Formalism/FormalismVsPeano/Peano/Waismann: Formalism does not share Peano's assumption that we already know the meaning of the words "zero", "number", "successor". For the formalists the axioms are links of meaningless signs whose structure alone interests them. The symbols can then be interpreted in infinitely many ways, Russell e.g. suppose,
1. "0" should be 100 and "number" should be the numbers from 100 upwards. Then our principles are satisfied. Even the fourth applies: although 100 is the successor to 99, 99 is not a "number" in the newly defined sense. Can be done with any arbitrary number instead of 100.
2. "0" usual meaning: "number" is an "even number" "successor" should be a number that results from it by addition of 2.
: 0,2,4,6,8...
all 5 axioms by Peano are satisfied.
I 75
3. "0" should be the number 1, "number" should mean the series
1, 1/2, 1/4, 1/8, ...
and "successor" shall mean "half of". For the resulting series all 5 peanotic axioms apply.
They therefore do not characterize the notion of the series of numbers, but rather that of progession.
One could then understand under numerical terms some things that satisfy the axioms (Russell) WaismannVs: unsatisfying, e.g. we would then have no more possibility to distinguish the statement "There are 5 regular bodies" from the statement: "There are 105 regular bodies".
Were the axioms restricted by additions so as to give a complete characterization of the cardinal numbers?
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Löwenheim-Skolem has thwarted this hope.