MATH-lec 14 Flashcards
the basic peano axioms
The Basic Peano Axioms
1 0 is not the successor of any natural number (there is no x such that 0 =
Sx)
2 The successor relation is 1-1.
3 For all x, x + 0 = x
4 For all x and all y (x + Sy = S(x + y))
5 For all x, x ́ 0 = 0
6 For all x and all y (x ́ Sy = (x ́ y) + x)
Peano Arithmetic Proof of ‘1 + 2 = 3’:
1 “x(x + 0 = x) (Axiom)
2 “x”y(x + Sy = S(x + y)) (Axiom)
3 S0 + 0 = S0 (from 1 by UI)
4 “y(S0 + Sy = S(S0 + y)) (from 2 by UI)
5 S0 + S0 = S(S0 + 0) (from 4 by UI)
6 S0 + S0 = SS0 (from 3, 5 by Substitution)
7 S0 + SS0 = S(S0 +S0) (from 4 by UI)
The non-uniqueness objection to the claim that numbers are abstract objects
2.a Alternative set-theoretic accounts of the natural numbers
Consider the following two series of sets, and the accompanying definition of the ‘successor’
relation, and the numbers ‘0’, ‘1’, ‘2’:
Series 1 (The ‘successor’ of n = the set containing n and its predecessors in the series)
Æ, {Æ}, {Æ, {Æ}}, {Æ, {Æ}, {Æ, {Æ}}},…
0 = Æ; 1 = {Æ}; 2 = {Æ, {Æ}}
Series 2 (The ‘successor’ of n = the set containing n)
Æ, {Æ}, {{Æ}}, {{{Æ}}},…
0 = Æ; 1 = {Æ}; 2 = {{Æ}}
The Peano Axioms can be derived using either the Series 1 approach or the series 2 approach.
Both approaches get us
- The first member of the series (zero) is not a successor of any natural number. (Ax1)
- Each natural number has exactly one successor; each natural number greater than zero is the successor of exactly
one natural number. (Ax2)
….
And either series can be used to count sets of things out in the world:
- on the Series 1 approach, a set has n members iff it can be put in 1-1 correspondence with n.
- on the Series 2 approach, a set has n members iff it can be put in 1-1 correspondence with
{x: x is a number less than n}
What numbers could not be
he existence of alternative set-theoretic constructions of the natural numbers generates an
argument against the claim that numbers are objects:
1 The Peano Axioms are derivable under both the Series 1 and the Series 2 set-theoretic accounts
of the natural number series. And both accounts can be used to count things out in the world.
2 There are no further constraints on what a natural number is than that it generate the Peano
Axioms and enable us to count things out in the world.
3 If numbers were objects, there would be a single right account of which objects they are.
So
4 Numbers are not objects.
The laws of identity
Given the laws of identity, we can’t say that both accounts of the natural numbers are right: this
would be to say that (for example) 2 = {Æ, {Æ}} and 2 = {{Æ}}. But the two sets have different
properties. So (applying the Second Law) we would end up with 2 ≠ 2 (which contradicts the
First Law).
The objection to the very idea of an abstract object
The argument in §2 grants that there are (or could be) abstract objects, but maintains that
numbers should not be identified with them. The second metaphysical argument against
Platonism is an argument for the conclusion that there are not any abstract objects at all:
1 The concept object is a category concept whose application is fixed by paradigm instances and
relations of relevant similarity: the paradigms are ordinary material things; something that is not
an ordinary material thing counts as an ‘object’ iff it is relevantly similar to things of this basic
kind.
2 The defining feature of an ordinary material thing is causal unity at a time and over time:
- causal unity at a time: changes in one part of an ordinary material thing impact causally on
other parts.
- causal unity over time: what a thing is like at t is causally dependent on what it is like at t-1,
and contributes to causally determining what it is like at t+1.
3 If x does not resemble paradigm Y’s with respect to their defining features, x is not a member
of the category of Y’s.
4 The notions of causal unity at a time and over time have no application to abstract entities.
So
5 No abstract entity is an object. (Or, putting this another way, the term ‘abstract object’ is
incoherent.)
