Logicism

Things from Last Time

Axiom of Regularity

( x)[(Ǝa)(a ∀ ϵ x) → (Ǝy)(y ϵ x & ~(Ǝz)(z ϵ x & z ϵ y))]

If you have a set xAnd x is not emptyThen one of x’s members yShares no members in common with x.

Doesn’t Allow S ϵ S

( x)[(Ǝa)(a ∀ ϵ x) → (Ǝy)(y ϵ x & ~(Ǝz)(z ϵ x & z ϵ y))]

Consider a set S– any set.S ϵ {S}So by Regularity, one of {S}’s members– that means S– has no members in common with {S}That is: nothing in S is in {S}So in particular, NOT: S ϵ S

Doesn’t Solve Russell’s Paradox

Regularity says that no set is a member of itself.

That by itself doesn’t allow or disallow the set of all non-self-membered sets.

It just says IF there is such a set, then it’s the set of all sets.

How to Solve Russell’s Paradox: A GuideT0 Ø exists.T1 Ø, { Ø } exist.T2 Ø, { Ø }, { { Ø } }, { Ø, { Ø } } exist.T3 Ø, { Ø }, { { Ø } }, { Ø, { Ø } }, POW[ Ø, {Ø}, {{Ø}}, {Ø, {Ø}} ] exist.T4 = T3 POW[ T3 ] ∪…

Just make sure to put anything contradictory on the list. (Notice the set of all sets never shows up.)

Set of All Sets Paradox

Russell’s paradox comes back if we allow a set of all sets.

NC: For every predicate F: (Ǝy)( x)(x ∀ ϵ y ↔ Fx)

RC: For every predicate F: ( z)(Ǝy)( x)(x ∀ ∀ ϵ y ↔ Fx & x ϵ z)

But NC = RC in the special case where z is the set of all sets.

Smaller and Larger Infinities

Cantor’s Diagonal Proof

Numbers vs. Numerals

Decimal Representations

A decimal representation of a real number consists of two parts:

A finite string S1 of Arabic numerals.

An infinite string S2 of Arabic numerals.

It looks like this:

S1 . S2

We can’t actually write out any decimal representations, since we can’t write infinite strings of numerals.

But we can write out abbreviations of some decimal representations.

1/4 = 0.251/7 = 0.142857

π = ?

_______

We will prove that there cannot be a list of all the decimal representations between ‘0.0’ and ‘1.0’.

A list is something with a first member, then a second member, then a third member and so on, perhaps continuing forever.

Choose an Arbitrary List

1. ‘8’ ‘4’ ‘3’ ‘0’ ‘0’ ‘0’ ‘0’ ‘0’ …

2. ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ …

3. ‘7’ ‘9’ ‘2’ ‘5’ ‘1’ ‘0’ ‘7’ ‘2’ …

4. ‘9’ ‘8’ ‘0’ ‘6’ ‘4’ ‘2’ ‘8’ ‘1’ …

5. ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ …

6. ‘4’ ‘3’ ‘7’ ‘7’ ‘1’ ‘0’ ‘2’ ‘0’ …

7. ‘8’ ‘8’ ‘1’ ‘3’ ‘2’ ‘9’ ‘9’ ‘6’ …

8. ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ …

…

Find the Diagonal

1. ‘8’ ‘4’ ‘3’ ‘0’ ‘0’ ‘0’ ‘0’ ‘0’ …

2. ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ …

3. ‘7’ ‘9’ ‘2’ ‘5’ ‘1’ ‘0’ ‘7’ ‘2’ …

4. ‘9’ ‘8’ ‘0’ ‘6’ ‘4’ ‘2’ ‘8’ ‘1’ …

5. ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ …

6. ‘4’ ‘3’ ‘7’ ‘7’ ‘1’ ‘0’ ‘2’ ‘0’ …

7. ‘8’ ‘8’ ‘1’ ‘3’ ‘2’ ‘9’ ‘9’ ‘6’ …

8. ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ …

…

Diagonal = 0.85263096…

Add move each numeral ‘1 up’– so ‘8’ becomes ‘9’, ‘5’ becomes ‘6’, etc.

New Representation = 0.96374107…

New Number Not on the List

‘9’ ‘6’ ‘3’ ‘7’ ‘4’ ‘1’ ‘0’ ‘7’ …

1. ‘8’ ‘4’ ‘3’ ‘0’ ‘0’ ‘0’ ‘0’ ‘0’ …

2. ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ …

3. ‘7’ ‘9’ ‘2’ ‘5’ ‘1’ ‘0’ ‘7’ ‘2’ …

4. ‘9’ ‘8’ ‘0’ ‘6’ ‘4’ ‘2’ ‘8’ ‘1’ …

5. ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ …

6. ‘4’ ‘3’ ‘7’ ‘7’ ‘1’ ‘0’ ‘2’ ‘0’ …

7. ‘8’ ‘8’ ‘1’ ‘3’ ‘2’ ‘9’ ‘9’ ‘6’ …

8. ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ …

…

Doesn’t Help to Add It In!

‘9’ ‘6’ ‘3’ ‘7’ ‘4’ ‘1’ ‘0’ ‘7’ …

1. ‘8’ ‘4’ ‘3’ ‘0’ ‘0’ ‘0’ ‘0’ ‘0’ …

2. ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ ‘6’ ‘2’ ‘5’ …

3. ‘7’ ‘9’ ‘2’ ‘5’ ‘1’ ‘0’ ‘7’ ‘2’ …

4. ‘9’ ‘8’ ‘0’ ‘6’ ‘4’ ‘2’ ‘8’ ‘1’ …

5. ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ ‘3’ …

6. ‘4’ ‘3’ ‘7’ ‘7’ ‘1’ ‘0’ ‘2’ ‘0’ …

7. ‘8’ ‘8’ ‘1’ ‘3’ ‘2’ ‘9’ ‘9’ ‘6’ …

8. ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ ‘1’ ‘6’ …

…

Discussion Questions

• Does this prove you can’t list all the real numbers?• How do we fix the proof?• Can you use a similar proof to show that the rational numbers aren’t

countable?• Can you list the powerset of the natural numbers?

Logicism

Immanuel Kant

• German philosopher• 1724-1804• Usually in the top ~3 of Western

philosophers• Most important work: Critique of

Pure Reason• Argued that arithmetical truths

are synthetic a priori

Analytic/ Synthetic Distinction

• Examples of analytic truths: geometric terms, kinship terms, animal terms (boar, sow, piglet, drift, pork)• True in virtue of meaning?• Relation to definitions?• Synthetic truths: “truth depends on actual facts”

A priori/ A posteriori

• Epistemological distinction• A priori “can be known prior to the experience of facts”

A Priori Knowledge

A priori/ A posteriori

• Epistemological distinction• A priori “can be known prior to the experience of facts” • Examples: analytic truths• Some experience necessary: learning the concepts• Other examples: Cartesian truths• “I exist” true in virtue of meaning?• A priori known vs. knowable: computing sums w/ calculator• A posteriori “can only be known as a result of relevant experiences”

Synthetic A Priori?

• Truth depends on actual facts, not just word meanings/ can be known without investigating actual facts• Examples? “All triangles have interior angles that sum to π radians”?• “The real numbers can’t be paired one-to-one with the integers”?• “The future will resemble the past”?• Universal Grammar?

The Knowledge of Babies

“In a few domains, babies seem to have intuitions that guide their expectations about how important entities in the world (e.g., objects, people) act and interact. For example, babies appear to be born knowing that objects cannot magically appear or disappear, that they cannot pass through each other, and that they cannot move unless contacted by another object. These expectations hold for objects, but not for non-object entities like substances (e.g., liquid, sand).”

--Kristy vanMarle

Gottlob Frege

• German mathematician and philosopher• With Bertrand Russell, one of

the founding figures of “analytic philosophy”• Argued that mathematics was

reducible to logic, and was thus analytic, not synthetic, a priori.

Frege-Analytic

Frege understood “being analytic” as being derivable from logic + definitions.

So his goal was to show that math is derivable from logic + definitions.

Set Theory

Central idea: the principles that give us sets are logical truths.

When there are some things, there is a collection of those things.

Discussion

• Is Frege’s conception of analyticity the right one?• On that conception, is it true that set theory is just logic?• Consider the null set

The von Neumann Construction of Arithmetic

The Structure of the Natural Numbers0 there’s a first one

The Structure of the Natural Numbers0 there’s a first ones0 then exactly one next one

The Structure of the Natural Numbers0 there’s a first ones0 then exactly one next oness0 then exactly one next one

The Structure of the Natural Numbers0 there’s a first ones0 then exactly one next oness0 then exactly one next onesss0ssss0sssss0ssssss0

and so on…

What We Need

• A set to identify as the first natural number, 0.• A definable successor function.• A relation that well-orders the sets in the range of the successor

function.

Some Thoughts

• It would be nice if Ø = 0• It would be nice if for every number N, the set that we identify with it

has N members.• It would be nice if the fundamental relation of set theory ϵ were

identified with the fundamental relation of arithmetic < .

Something Nice

0 = Ø1 = {0}2 = {0, 1}3 = {0, 1, 2}4 = {0, 1, 2, 3}5 = {0, 1, 2, 3, 4} And so on…

Something Nice

0 = Ø Ø1 = {0} { Ø }2 = {0, 1} { Ø, { Ø } }3 = {0, 1, 2} { Ø, { Ø }, { Ø, { Ø } } }4 = {0, 1, 2, 3} { Ø, { Ø }, { Ø, { Ø } }, {Ø, {Ø}, {Ø, {Ø}}} }5 = {0, 1, 2, 3, 4} And so on…

Successor Function

s(x) = x { x }∪

Addition

The Laws of Addition:

Commutativity: a + b = b + aAssociativity: (a + b) + c = a + (b + c)Identity Element: a + 0 = aSuccessor: a + 1 = s(a)

Recursive Definition of Addition

Base case: x + 0 = xRecursive step: x + s(y) = s(x + y)

Proof: 2 + 2 = 4

2 + 2= ss0 + ss0 By def. 2= s(ss0 + s0) By def. += ss(ss0 + 0) By def. += ss(ss0) By def. += ssss0= 4 By def. 4

The Zermelo Construction of the Natural Numbers

Zermelo’s Construction

0 = Ø1 = { 0 }2 = { { 0 } }3 = { { { 0 } } }4 = { { { { 0 } } }And so on… s(x) = { x }

Quick Evaluation

• 0 = Ø• Well-ordered by the ancestral of ϵ rather than ϵ itself• Doesn’t have the property that every number n has n members.

Benacerraf’s Problem

Paul Benacerraf

• American philosopher• Born in 1931• Teaching at Princeton since 1960• Argues against identifying

numbers with sets

Ernie and Johnny

• Ernie and Johnny are each raised by parents who believe that numbers are sets.• Each child is taught set theory

first, before they learn to count.• Then Ernie is taught von

Neumann’s construction of arithmetic and Johnny is taught Zermelo’s construction.

Ernie and Johnny

• All of the “pure set theory” each boy can prove will be the same, for example:• There is only one null set.• All of the “pure arithmetic” each

boy can prove will be the same, for example:• 2 + 2 = 4

But…

• For Ernie, 3 ϵ 17, but not for Johnny.• For Ernie, every set with 3 members has the same number of

members as the number 3, but not for Johnny.• For Ernie, 3 = { Ø, { Ø }, { Ø, { Ø } } }, but for Johnny, 3 = { { { Ø } } }

From “On What Numbers Could Not Be”

If “Is 3 = { { { Ø } } }?” “has an answer, there are arguments supporting it, and if there are no such arguments, then there is no ‘correct’ account that discriminates among [the different constructions of arithmetic in set theory]”

Frege’s View

Frege actually didn’t use either Zermelo’s or von Neumann’s construction.

For Frege 3 = the set of all 3-membered sets.

Frege’s View

3 =

{ x | Ǝy1Ǝy2Ǝy3

y1 ≠ y2 & y2 ≠ y3 & y3 ≠ y1

y1 ϵ x & y2 ϵ x & y3 ϵ x

& ~Ǝz (z ≠ y1 & z ≠ y2 & z ≠ y3

& z ϵ x) }

Argument for Frege’s View

• When we say “the sky is blue”, this is true iff the sky ϵ { x | x is blue }• When we say “Michael is smart”, this is true iff Michael ϵ { x | x is

smart }• So when we say “These lions are 3 (in number)” it should be true that

these lions (namely, the set of the lions) ϵ { x | x has 3 members }• And that’s Frege’s view!

Weak Argument

Benacerraf doesn’t think this is very plausible.

Numbers in language function more like what we call quantifiers:• All lions are in the zoo.• Some lions are in the zoo.• Five lions are in the zoo.

Finally, the set of all three membered sets leads to a paradox.

From “On What Numbers Could Not Be”

“There is no way connected with the reference of number words that will allow us to choose among them, for the accounts differ at places where there is no connection whatever between features of the accounts and our uses of the words in question.”

From “On What Numbers Could Not Be”

“[A]ny system of objects, whether sets or not, that forms a recursive progression must be adequate. But… any recursive set can be arranged in a recursive progression.”

From “On What Numbers Could Not Be”

[This] suggests that what is important is not the individuality of each element but the structure which they jointly exhibit… ‘Objects’ do not do the job of numbers singly; the whole system performs the job or nothing does.”

From “On What Numbers Could Not Be”“I therefore argue, extending the argument that led to the conclusion that numbers could not be sets, that numbers could not be objects at all; for there is no more reason to identify any individual number with any one particular object than with any other (not already known to be a number).”

From “On What Numbers Could Not Be”“To be the number 3 is no more and no less than to be preceded by 2, 1, and possibly 0, and to be followed by… Any object can play the role of 3; that is, any object can be the third element in some progression… [3 represents] the relation that any third member of a progression bears to the rest of the progression.”

From “On What Numbers Could Not Be”

“Arithmetic is therefore the science that elaborates the abstract structure that all progressions have in common merely in virtue of being progressions. It is not a science concerned with particular objects – the numbers.”

Discussion

• If this is right, what’s left of Kant’s claim that arithmetic is synthetic a priori?• If this is right, what’s left of logicism?• If this is right, should we say the same thing about sets? Is set theory

not the science of some objects – the sets?