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Single Idea 10704
[filed under theme 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
]
Full Idea
In second-order logic only the formation rules are completely formalizable, not the inference rules.
Gist of Idea
We can formalize second-order formation rules, but not inference rules
Source
Michael Potter (Set Theory and Its Philosophy [2004], 01.2)
Book Ref
Potter,Michael: 'Set Theory and Its Philosophy' [OUP 2004], p.13
A Reaction
He cites Gödel's First Incompleteness theorem for this.
The
14 ideas
from 'Set Theory and Its Philosophy'
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10702
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Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning
[Potter]
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10703
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Supposing axioms (rather than accepting them) give truths, but they are conditional
[Potter]
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10704
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We can formalize second-order formation rules, but not inference rules
[Potter]
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13041
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Collections have fixed members, but fusions can be carved in innumerable ways
[Potter]
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10707
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Mereology elides the distinction between the cards in a pack and the suits
[Potter]
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10708
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Nowadays we derive our conception of collections from the dependence between them
[Potter]
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13042
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If dependence is well-founded, with no infinite backward chains, this implies substances
[Potter]
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10709
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Priority is a modality, arising from collections and members
[Potter]
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10712
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If set theory didn't found mathematics, it is still needed to count infinite sets
[Potter]
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10713
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Usually the only reason given for accepting the empty set is convenience
[Potter]
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13043
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A relation is a set consisting entirely of ordered pairs
[Potter]
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13044
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Infinity: There is at least one limit level
[Potter]
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17882
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It is remarkable that all natural number arithmetic derives from just the Peano Axioms
[Potter]
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13546
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The 'limitation of size' principles say whether properties collectivise depends on the number of objects
[Potter]
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