Philosophical Devices
Week 3 – Ordering up Infinity
--------------------------------------------------------------------------------------------------------REVIEW: SIZING UP INFINITE SETS
 Cardinality: a measure of the number of elements of the set
We can associate with each finite set a natural number which represents its cardinality. Sets
with the same cardinality form an equivalence class.
Like finite sets, infinite sets can be grouped into equivalence classes, such that all the sets in a
given equivalence class have the same cardinality. The name for the cardinality of all sets
equivalent to ℕ is ‘0’

Denumerable: an infinite set is denumerable iff it can be mapped, one-to-one with ℕ
General strategy: To show that a set is denumerable, find a way to put it into a 1-1
correspondence with the natural numbers. This is called effectively listing the members of the
set, since the natural numbers themselves are thought of as forming an infinite ‘list’.
Problem: ℝ is non-denumerable - there are more real numbers then natural numbers. This
means that there is more than one equivalence class of infinite sets!
But how many such equivalence classes are there?
--------------------------------------------------------------------------------------------------------EQUIVALENCE CLASS 1: {REALS BETWEEN 0-1} = (ℕ)
Step 1: Re-write every real number as a binary expression.
In decimal notation, numbers to the left side of the decimal point increase in value by factors
of ten for each place moved over and numbers to right decrease in value by factors of ten.
Binary notation works similarly, except numbers to the left increase in value by factors of
two, while numbers to the right decrease by factors of two.
242.75 = 200 + 40 + 2 + 7/10 + 5/100 = 128 + 64 + 32 + 16 + 2 + 1/2 + 1/4 = 1111 0010.11
Unfortunately, only decimals that can be expressed as fractional powers of two can be
perfectly expressed in binary notation
0.21875 = 7/32 = 1/8 + 1/16 + 1/32 = 0.00111
0.375
= 3/8 = 1/4 + 1/8
= 0.011
0.1875 = 5/16 = 1/4 + 1/16
= 0.0101
To get the binary expression of a fraction, pull the fraction apart one piece at a time, then
reduce what's left until your result has a 1 in the numerator:
7/32 = 1/32 + 6/32
6/32 = 3/16 = 1/16 + 2/16
2/16 = 1/8
This gives us 1/8 + 1/16 + 1/32 or 0/2 + 0/4 + 1/8 + 1/16 + 1/32: 0.00111
Philosophical Devices
Week 3
2
Ordering up Infinites.
A direct technique: examine the fraction and see if we can pull a 1/2, a 1/4, a 1/8, etc. out of it
as we move along. For example, 0.21875 is smaller than 1/2, so we put a 0 in the first place,
smaller than 1/4, so we put a 0 in the second place, larger than 1/8, so we put a 1 in the third
place and pull out the 1/8 (0.125) from the fraction (leaving 0.09375). We do the same with
1/16 (leaving 0.03125) and 1/32 (leaving 0) to get our answer: 0.00111
7/32
7/32
7/32
7/32
3/32
1/32
<
<
<
>
>
=
1
1/2
1/4
1/8
1/16
1/32
(16/32)
(8/32)
(4/32)
(2/32)
→
→
→
→
→
→
0.
0
0
1 (remove 1/8 from the fraction)
1 (remove 1/16 from the fraction)
1 (finished)
Answer: 0.00111
You can simplify the algorithm if you multiply the fraction by 2 at each step in the process.
This means we can make a much simpler test, checking to see if the fraction greater than or
equal to 1, rather than is it greater than or equal to 1/2n. Multiplying the first step by 2 means
we test for 1 instead of 1/2, then multiplying by another 2 at step two means we test for 1
instead of 1/4, etc. When the fraction is greater than one, pull the one out and put a 1 in that
place after the point; and when the fraction is equal to one, the algorithm terminates:
2
2
2
2
×
×
×
×
5/16
5/8
1/4
1/2
=
=
=
=
5/16
5/8
5/4
1/2
1
<
<
>
<
=
1
1
1
1
1
→
→
→
→
→
0.
0
1 (remove 1, or 4/4, from the fraction)
0
1 (finished)
Answer: 0.0101
Step 2: Apply the following Recipe
-
n is in subset K iff there is a ‘1’ in the n+1 digit of the real number
So each real number is a recipe for generating a subset which is a member of (ℕ), and each
member of (ℕ) is a recipe for generating an accompanying real number.
This gives us our 1-1 mapping: every member of (ℕ) is mapped to a real number, every
real number to a member of (ℕ).
Papineau labels this cardinality ‘2infinity0’ – recall that, if X has n members, (X) has 2n
--------------------------------------------------------------------------------------------------------INFINITE INFINITIES?
- Claim: for all sets X, the cardinality of (X) > cardinality of X
Suppose that ℕ is has the same cardinality as (ℕ). We can therefore attempt to pair off each
element of ℕ with an element from the infinite set
(ℕ), so that no element from either infinite set
remains unpaired, e.g:
Given such a pairing, some numbers are paired
with subsets they are members of – e.g., 2 is paired
with {1, 2, 3}, which has 2 as a member. Call such
Philosophical Devices
Week 3
3
Ordering up Infinites.
numbers selfish.
Other numbers are paired with subsets they aren’t members of – e.g. 1 is paired with {4, 5}.
Call these numbers non-selfish.
Let R be the set of all non-selfish natural numbers. By definition, (ℕ) contains all sets of
natural numbers, so R(ℕ). Given our mapping assumption, it follows that R must be
paired with some natural number – call it r.
Question: is rR? If r is in R then r is selfish. And, if r is selfish, then r cannot be a member
of R. Further, if rR, then r is non-selfish. This means that r R! So rR entails rR, and
rR entails rR.
This is a contradiction: the natural number r must be either in R or not in R, yet neither is
possible. Therefore, there is no natural number which can be paired with R. We have
contradicted our original supposition regarding the 1-1 mapping.
Note: R may be empty – e.g. every natural number x is mapped to a set that contains x. Then,
every number maps to a nonempty set and no number maps to . But (ℕ)!
Conclusion: our claim is true – (2infinity0)> 2infinity0, ((2infinity0))> (2infinity0), etc…
--------------------------------------------------------------------------------------------------------CONTINUUM HYPOTHESIS
Is there an infinity between 0 and 2infinity0?
Cantor proposed that wasn’t (i.e. according to Cantor, 1 = 2infinity0). He didn’t give a proof.

-
Continuum Hypothesis: There is no set whose cardinality is strictly between
that of the integers and that of the real numbers
Gödel (1940) showed that no contradiction would arise if the continuum hypothesis
were added to conventional ZF or ZFC.
Cohen (1963, 1964) used forcing to prove that no contradiction arises if the negation
of the continuum hypothesis was added to ZF or ZFC
Conclusion: the Continuum Hypothesis is independent of ZF/ZFC. Maybe there is an infinity
in-between, maybe there isn’t – set theory itself doesn’t really settle the matter.
-
The list of ’s = the sequence of all infinite numbers, arranged in ascending order
Papineau’s 2infinity0, 22infinity0, = the sequence of power set cardinalities, in ascending
order

General Continuum Hypothesis: if an infinite set's cardinality lies between
that of an infinite set A and that of the power set of A, then it either has the
same cardinality as A or the same cardinality as (A)
GCH is the claim that there are no infinite numerical sizes between those generated by
repeatedly taking the power set of the natural numbers – in other words, it is the claim that
Papineau’s sequence and the list of ’s coincide – e.g. that 2infinity0 = 1, 22infinity0 = 2…