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Single Idea 13670
[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
]
Full Idea
Categoricity cannot be attained in a first-order language.
Gist of Idea
Categoricity can't be reached in a first-order language
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 7.3)
Book Ref
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.196
Related Ideas
Idea 10758
If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model [Rossberg]
Idea 13636
An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]
The
50 ideas
from 'Foundations without Foundationalism'
15944
|
Second-order logic is better than set theory, since it only adds relations and operations, and nothing else
[Shapiro, by Lavine]
|
13624
|
The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed
[Shapiro]
|
13629
|
Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics?
[Shapiro]
|
13631
|
Are sets part of logic, or part of mathematics?
[Shapiro]
|
13627
|
There is no 'correct' logic for natural languages
[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]
|
13626
|
Semantic consequence is ineffective in second-order logic
[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]
|
13638
|
Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects
[Shapiro]
|
13640
|
Russell's paradox shows that there are classes which are not iterative sets
[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]
|
13646
|
Compactness is derived from soundness and completeness
[Shapiro]
|
13648
|
The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity
[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]
|
13666
|
Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets
[Shapiro]
|
13662
|
First-order logic was an afterthought in the development of modern logic
[Shapiro]
|
13664
|
Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions
[Shapiro]
|
13663
|
Some reject formal properties if they are not defined, or defined impredicatively
[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]
|
13676
|
Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are
[Shapiro]
|
13674
|
We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models
[Shapiro]
|
13675
|
Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails
[Shapiro]
|
13677
|
Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals
[Shapiro]
|