more on this theme     |     more from this thinker


Single Idea 18102

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers ]

Full Idea

It is the usual procedure these days to identify a cardinal number with the earliest ordinal number that has that number of predecessors.

Gist of Idea

A cardinal is the earliest ordinal that has that number of predecessors

Source

David Bostock (Philosophy of Mathematics [2009], 4.5)

Book Ref

Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.111


A Reaction

This sounds circular, since you need to know the cardinal in order to decide which ordinal is the one you want, but, hey, what do I know?


The 49 ideas from 'Philosophy of Mathematics'

Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock]
For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock]
The Peano Axioms describe a unique structure [Bostock]
The number of reals is the number of subsets of the natural numbers [Bostock]
ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock]
Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock]
A cardinal is the earliest ordinal that has that number of predecessors [Bostock]
Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock]
A 'proper class' cannot be a member of anything [Bostock]
Replacement enforces a 'limitation of size' test for the existence of sets [Bostock]
The completeness of first-order logic implies its compactness [Bostock]
First-order logic is not decidable: there is no test of whether any formula is valid [Bostock]
Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock]
Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock]
Numbers can't be positions, if nothing decides what position a given number has [Bostock]
We could add axioms to make sets either as small or as large as possible [Bostock]
There is no single agreed structure for set theory [Bostock]
Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock]
Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock]
The Deduction Theorem is what licenses a system of natural deduction [Bostock]
In logic a proposition means the same when it is and when it is not asserted [Bostock]
Substitutional quantification is just standard if all objects in the domain have a name [Bostock]
Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock]
Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock]
If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock]
Many crucial logicist definitions are in fact impredicative [Bostock]
The predicativity restriction makes a difference with the real numbers [Bostock]
Impredicative definitions are wrong, because they change the set that is being defined? [Bostock]
Predicativism makes theories of huge cardinals impossible [Bostock]
If mathematics rests on science, predicativism may be the best approach [Bostock]
If we can only think of what we can describe, predicativism may be implied [Bostock]
The usual definitions of identity and of natural numbers are impredicative [Bostock]
The best version of conceptualism is predicativism [Bostock]
Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock]
The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock]
Nominalism about mathematics is either reductionist, or fictionalist [Bostock]
Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock]
Higher cardinalities in sets are just fairy stories [Bostock]
Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock]
Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock]
If Hume's Principle is the whole story, that implies structuralism [Bostock]
There are many criteria for the identity of numbers [Bostock]
Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock]
Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock]
Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock]
Actual measurement could never require the precision of the real numbers [Bostock]
A fairy tale may give predictions, but only a true theory can give explanations [Bostock]
Modern axioms of geometry do not need the real numbers [Bostock]
Nominalism as based on application of numbers is no good, because there are too many applications [Bostock]