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Single Idea 13652
[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
]
Full Idea
The 'continuum' is the cardinality of the powerset of a denumerably infinite set.
Gist of Idea
The 'continuum' is the cardinality of the powerset of a denumerably infinite set
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.2)
Book Ref
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.105
The
142 ideas
from Stewart Shapiro
15944
|
Second-order logic is better than set theory, since it only adds relations and operations, and nothing else
[Shapiro, by Lavine]
|
13629
|
Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics?
[Shapiro]
|
13627
|
There is no 'correct' logic for natural languages
[Shapiro]
|
13624
|
The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed
[Shapiro]
|
13631
|
Are sets part of logic, or part of mathematics?
[Shapiro]
|
13626
|
Semantic consequence is ineffective in second-order logic
[Shapiro]
|
13628
|
We can live well without completeness in logic
[Shapiro]
|
13630
|
Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures
[Shapiro]
|
13625
|
Mathematics and logic have no border, and logic must involve mathematics and its ontology
[Shapiro]
|
13632
|
Finding the logical form of a sentence is difficult, and there are no criteria of correctness
[Shapiro]
|
13633
|
'Satisfaction' is a function from models, assignments, and formulas to {true,false}
[Shapiro]
|
13635
|
'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence
[Shapiro]
|
13634
|
Satisfaction is 'truth in a model', which is a model of 'truth'
[Shapiro]
|
13636
|
An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation
[Shapiro]
|
13637
|
If a logic is incomplete, its semantic consequence relation is not effective
[Shapiro]
|
13640
|
Russell's paradox shows that there are classes which are not iterative sets
[Shapiro]
|
13638
|
Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects
[Shapiro]
|
13641
|
Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals
[Shapiro]
|
13642
|
Logic is the ideal for learning new propositions on the basis of others
[Shapiro]
|
13643
|
Aristotelian logic is complete
[Shapiro]
|
13644
|
Semantics for models uses set-theory
[Shapiro]
|
13650
|
Henkin semantics has separate variables ranging over the relations and over the functions
[Shapiro]
|
13645
|
In standard semantics for second-order logic, a single domain fixes the ranges for the variables
[Shapiro]
|
13649
|
Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics
[Shapiro]
|
13647
|
Choice is essential for proving downward Löwenheim-Skolem
[Shapiro]
|
13648
|
The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity
[Shapiro]
|
13646
|
Compactness is derived from soundness and completeness
[Shapiro]
|
13651
|
A set is 'transitive' if contains every member of each of its members
[Shapiro]
|
13657
|
First-order arithmetic can't even represent basic number theory
[Shapiro]
|
13652
|
The 'continuum' is the cardinality of the powerset of a denumerably infinite set
[Shapiro]
|
13653
|
'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element
[Shapiro]
|
13654
|
It is central to the iterative conception that membership is well-founded, with no infinite descending chains
[Shapiro]
|
13656
|
Some sets of natural numbers are definable in set-theory but not in arithmetic
[Shapiro]
|
13661
|
A language is 'semantically effective' if its logical truths are recursively enumerable
[Shapiro]
|
13660
|
Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable
[Shapiro]
|
13659
|
Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes
[Shapiro]
|
13658
|
Downward Löwenheim-Skolem: each satisfiable countable set always has countable models
[Shapiro]
|
13662
|
First-order logic was an afterthought in the development of modern logic
[Shapiro]
|
13666
|
Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets
[Shapiro]
|
13663
|
Some reject formal properties if they are not defined, or defined impredicatively
[Shapiro]
|
13664
|
Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions
[Shapiro]
|
13667
|
Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order
[Shapiro]
|
13668
|
Bernays (1918) formulated and proved the completeness of propositional logic
[Shapiro]
|
13669
|
Can one develop set theory first, then derive numbers, or are numbers more basic?
[Shapiro]
|
13670
|
Categoricity can't be reached in a first-order language
[Shapiro]
|
13673
|
The notion of finitude is actually built into first-order languages
[Shapiro]
|
13675
|
Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails
[Shapiro]
|
13674
|
We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models
[Shapiro]
|
13676
|
Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are
[Shapiro]
|
13677
|
Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals
[Shapiro]
|
10290
|
Second-order variables also range over properties, sets, relations or functions
[Shapiro]
|
10590
|
Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them
[Shapiro]
|
10292
|
Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model
[Shapiro]
|
10588
|
First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems
[Shapiro]
|
10294
|
Second-order logic has the expressive power for mathematics, but an unworkable model theory
[Shapiro]
|
10591
|
Logicians use 'property' and 'set' interchangeably, with little hanging on it
[Shapiro]
|
10296
|
The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics
[Shapiro]
|
10297
|
The Löwenheim-Skolem theorem seems to be a defect of first-order logic
[Shapiro]
|
10300
|
Logical consequence can be defined in terms of the logical terminology
[Shapiro]
|
10298
|
Some say that second-order logic is mathematics, not logic
[Shapiro]
|
10299
|
If the aim of logic is to codify inferences, second-order logic is useless
[Shapiro]
|
10301
|
The axiom of choice is controversial, but it could be replaced
[Shapiro]
|
10279
|
Can we discover whether a deck is fifty-two cards, or a person is time-slices or molecules?
[Shapiro]
|
10203
|
We apprehend small, finite mathematical structures by abstraction from patterns
[Shapiro]
|
10200
|
We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false)
[Shapiro]
|
10205
|
Mathematics originally concerned the continuous (geometry) and the discrete (arithmetic)
[Shapiro]
|
10201
|
Virtually all of mathematics can be modeled in set theory
[Shapiro]
|
10202
|
Natural numbers just need an initial object, successors, and an induction principle
[Shapiro]
|
10204
|
An 'implicit definition' gives a direct description of the relations of an entity
[Shapiro]
|
10206
|
Modal operators are usually treated as quantifiers
[Shapiro]
|
10207
|
Anti-realists reject set theory
[Shapiro]
|
10208
|
Axiom of Choice: some function has a value for every set in a given set
[Shapiro]
|
10209
|
A function is just an arbitrary correspondence between collections
[Shapiro]
|
10210
|
If mathematical objects are accepted, then a number of standard principles will follow
[Shapiro]
|
10213
|
Real numbers are thought of as either Cauchy sequences or Dedekind cuts
[Shapiro]
|
10214
|
Theory ontology is never complete, but is only determined 'up to isomorphism'
[Shapiro]
|
10212
|
Classical connectives differ from their ordinary language counterparts; '∧' is timeless, unlike 'and'
[Shapiro]
|
10218
|
Baseball positions and chess pieces depend entirely on context
[Shapiro]
|
10215
|
Platonists claim we can state the essence of a number without reference to the others
[Shapiro]
|
10217
|
We can apprehend structures by focusing on or ignoring features of patterns
[Shapiro]
|
10220
|
Because one structure exemplifies several systems, a structure is a one-over-many
[Shapiro]
|
10222
|
Mathematical foundations may not be sets; categories are a popular rival
[Shapiro]
|
10221
|
Is there is no more to structures than the systems that exemplify them?
[Shapiro]
|
10223
|
There is no 'structure of all structures', just as there is no set of all sets
[Shapiro]
|
10224
|
The even numbers have the natural-number structure, with 6 playing the role of 3
[Shapiro]
|
10228
|
Could infinite structures be apprehended by pattern recognition?
[Shapiro]
|
10227
|
The abstract/concrete boundary now seems blurred, and would need a defence
[Shapiro]
|
10226
|
Mathematicians regard arithmetic as concrete, and group theory as abstract
[Shapiro]
|
10229
|
Simple types can be apprehended through their tokens, via abstraction
[Shapiro]
|
10230
|
The 4-pattern is the structure common to all collections of four objects
[Shapiro]
|
8703
|
Shapiro's structuralism says model theory (comparing structures) is the essence of mathematics
[Shapiro, by Friend]
|
10231
|
Abstract objects might come by abstraction over an equivalence class of base entities
[Shapiro]
|
10233
|
Platonism must accept that the Peano Axioms could all be false
[Shapiro]
|
10236
|
There is no grounding for mathematics that is more secure than mathematics
[Shapiro]
|
10237
|
Coherence is a primitive, intuitive notion, not reduced to something formal
[Shapiro]
|
10235
|
A sentence is 'satisfiable' if it has a model
[Shapiro]
|
10238
|
The set-theoretical hierarchy contains as many isomorphism types as possible
[Shapiro]
|
10234
|
Any theory with an infinite model has a model of every infinite cardinality
[Shapiro]
|
10239
|
The central notion of model theory is the relation of 'satisfaction'
[Shapiro]
|
10240
|
Model theory deals with relations, reference and extensions
[Shapiro]
|
18243
|
Understanding the real-number structure is knowing usage of the axiomatic language of analysis
[Shapiro]
|
10244
|
Intuition is an outright hindrance to five-dimensional geometry
[Shapiro]
|
10248
|
Number statements are generalizations about number sequences, and are bound variables
[Shapiro]
|
18245
|
Cuts are made by the smallest upper or largest lower number, some of them not rational
[Shapiro]
|
10249
|
The main mathematical structures are algebraic, ordered, and topological
[Shapiro]
|
10252
|
The Axiom of Choice seems to license an infinite amount of choosing
[Shapiro]
|
10251
|
The law of excluded middle might be seen as a principle of omniscience
[Shapiro]
|
10253
|
Either logic determines objects, or objects determine logic, or they are separate
[Shapiro]
|
10254
|
Can the ideal constructor also destroy objects?
[Shapiro]
|
10255
|
Presumably nothing can block a possible dynamic operation?
[Shapiro]
|
10256
|
For intuitionists, proof is inherently informal
[Shapiro]
|
10257
|
Intuitionism only sanctions modus ponens if all three components are proved
[Shapiro]
|
10258
|
Logical modalities may be acceptable, because they are reducible to satisfaction in models
[Shapiro]
|
10262
|
Fictionalism eschews the abstract, but it still needs the possible (without model theory)
[Shapiro]
|
10259
|
The two standard explanations of consequence are semantic (in models) and deductive
[Shapiro]
|
10268
|
Maybe plural quantifiers should be understood in terms of classes or sets
[Shapiro]
|
10266
|
Why does the 'myth' of possible worlds produce correct modal logic?
[Shapiro]
|
10270
|
The main versions of structuralism are all definitionally equivalent
[Shapiro]
|
10272
|
The notion of 'object' is at least partially structural and mathematical
[Shapiro]
|
10274
|
Does someone using small numbers really need to know the infinite structure of arithmetic?
[Shapiro]
|
10273
|
Some structures are exemplified by both abstract and concrete
[Shapiro]
|
10276
|
Mathematical structures are defined by axioms, or in set theory
[Shapiro]
|
10275
|
A blurry border is still a border
[Shapiro]
|
10277
|
Structuralism blurs the distinction between mathematical and ordinary objects
[Shapiro]
|
10280
|
A stone is a position in some pattern, and can be viewed as an object, or as a location
[Shapiro]
|
9554
|
We can focus on relations between objects (like baseballers), ignoring their other features
[Shapiro]
|
9626
|
A structure is an abstraction, focussing on relationships, and ignoring other features
[Shapiro]
|
8725
|
Rationalism tries to apply mathematical methodology to all of knowledge
[Shapiro]
|
8730
|
'Impredicative' definitions refer to the thing being described
[Shapiro]
|
8729
|
Intuitionists deny excluded middle, because it is committed to transcendent truth or objects
[Shapiro]
|
8731
|
Conceptualist are just realists or idealist or nominalists, depending on their view of concepts
[Shapiro]
|
8744
|
Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own
[Shapiro]
|
8749
|
Term Formalism says mathematics is just about symbols - but real numbers have no names
[Shapiro]
|
8750
|
Game Formalism is just a matter of rules, like chess - but then why is it useful in science?
[Shapiro]
|
8752
|
Deductivism says mathematics is logical consequences of uninterpreted axioms
[Shapiro]
|
8753
|
Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions
[Shapiro]
|
8760
|
Numbers do not exist independently; the essence of a number is its relations to other numbers
[Shapiro]
|
8761
|
A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them
[Shapiro]
|
8762
|
Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3
[Shapiro]
|
8763
|
The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex
[Shapiro]
|
8764
|
Categories are the best foundation for mathematics
[Shapiro]
|
18249
|
Cauchy gave a formal definition of a converging sequence.
[Shapiro]
|