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Single Idea 13023
[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
]
Full Idea
The Power Set Axiom is indispensable for a set-theoretic account of the continuum, ...and in so far as those attempts are successful, then the power-set principle gains some confirmatory support.
Gist of Idea
The Power Set Axiom is needed for, and supported by, accounts of the continuum
Source
Penelope Maddy (Believing the Axioms I [1988], §1.6)
Book Ref
-: 'Journal of Symbolic Logic' [-], p.486
A Reaction
The continuum is, of course, notoriously problematic. Have we created an extra problem in our attempts at solving the first one?
The
57 ideas
from Penelope Maddy
13011
|
New axioms are being sought, to determine the size of the continuum
[Maddy]
|
13013
|
The Axiom of Extensionality seems to be analytic
[Maddy]
|
13014
|
Extensional sets are clearer, simpler, unique and expressive
[Maddy]
|
13019
|
The Iterative Conception says everything appears at a stage, derived from the preceding appearances
[Maddy]
|
13018
|
Limitation of Size is a vague intuition that over-large sets may generate paradoxes
[Maddy]
|
13021
|
The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics
[Maddy]
|
13022
|
Infinite sets are essential for giving an account of the real numbers
[Maddy]
|
13023
|
The Power Set Axiom is needed for, and supported by, accounts of the continuum
[Maddy]
|
13024
|
Efforts to prove the Axiom of Choice have failed
[Maddy]
|
13025
|
Modern views say the Choice set exists, even if it can't be constructed
[Maddy]
|
13026
|
A large array of theorems depend on the Axiom of Choice
[Maddy]
|
17610
|
The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres
[Maddy]
|
17605
|
Hilbert's geometry and Dedekind's real numbers were role models for axiomatization
[Maddy]
|
17614
|
The connection of arithmetic to perception has been idealised away in modern infinitary mathematics
[Maddy]
|
17615
|
Every infinite set of reals is either countable or of the same size as the full set of reals
[Maddy]
|
17620
|
Critics of if-thenism say that not all starting points, even consistent ones, are worth studying
[Maddy]
|
17618
|
Set-theory tracks the contours of mathematical depth and fruitfulness
[Maddy]
|
17625
|
If two mathematical themes coincide, that suggest a single deep truth
[Maddy]
|
18182
|
The extension of concepts is not important to me
[Maddy]
|
18163
|
Mathematics rests on the logic of proofs, and on the set theoretic axioms
[Maddy]
|
18168
|
'Propositional functions' are propositions with a variable as subject or predicate
[Maddy]
|
18172
|
Infinity has degrees, and large cardinals are the heart of set theory
[Maddy]
|
18175
|
For any cardinal there is always a larger one (so there is no set of all sets)
[Maddy]
|
18169
|
Axiom of Reducibility: propositional functions are extensionally predicative
[Maddy]
|
18167
|
We can get arithmetic directly from HP; Law V was used to get HP from the definition of number
[Maddy]
|
18164
|
Frege solves the Caesar problem by explicitly defining each number
[Maddy]
|
18171
|
Cantor and Dedekind brought completed infinities into mathematics
[Maddy]
|
18177
|
In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets
[Maddy]
|
18187
|
Theorems about limits could only be proved once the real numbers were understood
[Maddy]
|
18186
|
Identifying geometric points with real numbers revealed the power of set theory
[Maddy]
|
18188
|
The line of rationals has gaps, but set theory provided an ordered continuum
[Maddy]
|
18184
|
Making set theory foundational to mathematics leads to very fruitful axioms
[Maddy]
|
18185
|
Unified set theory gives a final court of appeal for mathematics
[Maddy]
|
18183
|
Set theory brings mathematics into one arena, where interrelations become clearer
[Maddy]
|
18190
|
Completed infinities resulted from giving foundations to calculus
[Maddy]
|
18193
|
The Axiom of Foundation says every set exists at a level in the set hierarchy
[Maddy]
|
18191
|
Axiom of Infinity: completed infinite collections can be treated mathematically
[Maddy]
|
18194
|
'Forcing' can produce new models of ZFC from old models
[Maddy]
|
18195
|
A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy
[Maddy]
|
18196
|
An 'inaccessible' cardinal cannot be reached by union sets or power sets
[Maddy]
|
18204
|
Scientists posit as few entities as possible, but set theorist posit as many as possible
[Maddy]
|
18205
|
The theoretical indispensability of atoms did not at first convince scientists that they were real
[Maddy]
|
18206
|
Science idealises the earth's surface, the oceans, continuities, and liquids
[Maddy]
|
18207
|
Maybe applications of continuum mathematics are all idealisations
[Maddy]
|
8755
|
Maddy replaces pure sets with just objects and perceived sets of objects
[Maddy, by Shapiro]
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8756
|
Intuition doesn't support much mathematics, and we should question its reliability
[Maddy, by Shapiro]
|
17733
|
We know mind-independent mathematical truths through sets, which rest on experience
[Maddy, by Jenkins]
|
10718
|
A natural number is a property of sets
[Maddy, by Oliver]
|
10594
|
Henkin semantics is more plausible for plural logic than for second-order logic
[Maddy]
|
17823
|
If mathematical objects exist, how can we know them, and which objects are they?
[Maddy]
|
17825
|
Set theory (unlike the Peano postulates) can explain why multiplication is commutative
[Maddy]
|
17824
|
The master science is physical objects divided into sets
[Maddy]
|
17826
|
Standardly, numbers are said to be sets, which is neat ontology and epistemology
[Maddy]
|
17828
|
Numbers are properties of sets, just as lengths are properties of physical objects
[Maddy]
|
17827
|
Sets exist where their elements are, but numbers are more like universals
[Maddy]
|
17829
|
Number words are unusual as adjectives; we don't say 'is five', and numbers always come first
[Maddy]
|
17830
|
Number theory doesn't 'reduce' to set theory, because sets have number properties
[Maddy]
|