more on this theme
|
more from this text
Single Idea 9613
[filed under theme 4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
]
Full Idea
In the early versions of set theory ('naïve' set theory), the axiom of comprehension assumed that for any condition there is a set of objects satisfying that condition (so P(x)↔x∈{x:P(x)}), but this led directly to Russell's Paradox.
Clarification
See Idea 6407 for Russell's Paradox
Gist of Idea
Naïve set theory assumed that there is a set for every condition
Source
James Robert Brown (Philosophy of Mathematics [1999], Ch. 2)
Book Ref
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.19
A Reaction
How rarely any philosophers state this problem clearly (as Brown does here). This is incredibly important for our understanding of how we classify the world. I'm tempted to just ignore Russell, and treat sets in a natural and sensible way.
Related Idea
Idea 6407
The class of classes which lack self-membership leads to a contradiction [Russell, by Grayling]
The
33 ideas
from James Robert Brown
9605
|
If a proposition is false, then its negation is true
[Brown,JR]
|
9604
|
Mathematics is the only place where we are sure we are right
[Brown,JR]
|
9606
|
The irrationality of root-2 was achieved by intellect, not experience
[Brown,JR]
|
9612
|
There is an infinity of mathematical objects, so they can't be physical
[Brown,JR]
|
9610
|
Numbers are not abstracted from particulars, because each number is a particular
[Brown,JR]
|
9608
|
There are no constructions for many highly desirable results in mathematics
[Brown,JR]
|
9611
|
'Abstract' nowadays means outside space and time, not concrete, not physical
[Brown,JR]
|
9609
|
The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars
[Brown,JR]
|
9613
|
Naïve set theory assumed that there is a set for every condition
[Brown,JR]
|
9615
|
Nowadays conditions are only defined on existing sets
[Brown,JR]
|
9617
|
The 'iterative' view says sets start with the empty set and build up
[Brown,JR]
|
9619
|
David's 'Napoleon' is about something concrete and something abstract
[Brown,JR]
|
9620
|
Empiricists base numbers on objects, Platonists base them on properties
[Brown,JR]
|
9625
|
To see a structure in something, we must already have the idea of the structure
[Brown,JR]
|
9628
|
Sets seem basic to mathematics, but they don't suit structuralism
[Brown,JR]
|
9622
|
'There are two apples' can be expressed logically, with no mention of numbers
[Brown,JR]
|
9621
|
Mathematics represents the world through structurally similar models.
[Brown,JR]
|
9630
|
The most brilliant formalist was Hilbert
[Brown,JR]
|
9638
|
Berry's Paradox finds a contradiction in the naming of huge numbers
[Brown,JR]
|
9634
|
Set theory says that natural numbers are an actual infinity (to accommodate their powerset)
[Brown,JR]
|
9629
|
For nomalists there are no numbers, only numerals
[Brown,JR]
|
9635
|
Given atomism at one end, and a finite universe at the other, there are no physical infinities
[Brown,JR]
|
9640
|
A term can have not only a sense and a reference, but also a 'computational role'
[Brown,JR]
|
9639
|
Does some mathematics depend entirely on notation?
[Brown,JR]
|
9643
|
Set theory may represent all of mathematics, without actually being mathematics
[Brown,JR]
|
9644
|
When graphs are defined set-theoretically, that won't cover unlabelled graphs
[Brown,JR]
|
9641
|
Definitions should be replaceable by primitives, and should not be creative
[Brown,JR]
|
9642
|
A flock of birds is not a set, because a set cannot go anywhere
[Brown,JR]
|
9645
|
Constructivists say p has no value, if the value depends on Goldbach's Conjecture
[Brown,JR]
|
9646
|
There is no limit to how many ways something can be proved in mathematics
[Brown,JR]
|
9647
|
Computers played an essential role in proving the four-colour theorem of maps
[Brown,JR]
|
9649
|
Axioms are either self-evident, or stipulations, or fallible attempts
[Brown,JR]
|
9648
|
π is a 'transcendental' number, because it is not the solution of an equation
[Brown,JR]
|