HOUSE_OVERSIGHT_015845.jpg
1.42 MB
Extraction Summary
3
People
1
Organizations
0
Locations
1
Events
1
Relationships
4
Quotes
Document Information
Type:
Book page / manuscript / evidence document
File Size:
1.42 MB
Summary
This document appears to be page 155 from a book or manuscript titled 'Kittens & Gorillas', stamped with a House Oversight Bates number (015845), indicating it is part of a congressional investigation file. The text discusses mathematical logic, specifically the Barber Paradox and Russell's Paradox, referencing historical figures Bertrand Russell and Gottlob Frege. While the content is academic, its presence in this specific discovery collection suggests it may have been seized from a personal library or document cache relevant to the investigation.
People (3)
| Name | Role | Context |
|---|---|---|
| Bertrand Russell | Mathematician/Philosopher |
Mentioned as changing the course of mathematical history; wrote to Gottlob Frege pointing out a paradox.
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| Gottlob Frege | Mathematician |
Completed theory of sets in 1890; received letter from Russell regarding the paradox.
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| The Barber | Hypothetical subject |
Subject of the 'Barber Paradox' example.
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Organizations (1)
| Name | Type | Context |
|---|---|---|
| House Oversight Committee |
Implied by the Bates stamp 'HOUSE_OVERSIGHT' at the bottom of the page.
|
Timeline (1 events)
Relationships (1)
Bertrand Russell wrote to him and pointed out the following paradox.
Key Quotes (4)
"who shaves the barber?"Source
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Quote #1
"What about the set of sets that does not contain itself?"Source
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Quote #2
"The 'idea' of a number is to say something about the things I have in my hand"Source
HOUSE_OVERSIGHT_015845.jpg
Quote #3
"Bertrand Russell changed the course of mathematical history and it is the fundamental paradox used to show computers are Turing limited."Source
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Quote #4
Full Extracted Text
Complete text extracted from the document (2,194 characters)
Kittens & Gorillas
155
155
The mathematical reason for it being a false paradox is that some series converge and some do not. If I move progressively closer and closer to something in smaller and smaller time intervals then I may indeed reach it. On the other hand, some series never converge. I will never reach infinity how ever many steps I take.
The Barber Paradox
Now, for a slightly harder paradox, let’s suppose there is a town with just one barber.
In this town, every man keeps himself clean-shaven by either shaving himself or going to the barber; the barber shaves all the men in town who do not shave themselves. All this seems perfectly logical, until we pose the question: who shaves the barber?
This question results in a paradox because, according to the statement above, he can either be shaven by himself or the barber, which is he. However, neither of these possibilities is valid! This is because if the barber shaves himself, then the barber must not shave himself and if the barber does not shave himself, then the barber must shave himself.
You might think this paradox an oddity but, using this simple idea, Bertrand Russell changed the course of mathematical history and it is the fundamental paradox used to show computers are Turing limited.
Now, for a slightly harder paradox, let’s suppose there is a town with just one barber.
In this town, every man keeps himself clean-shaven by either shaving himself or going to the barber; the barber shaves all the men in town who do not shave themselves. All this seems perfectly logical, until we pose the question: who shaves the barber?
This question results in a paradox because, according to the statement above, he can either be shaven by himself or the barber, which is he. However, neither of these possibilities is valid! This is because if the barber shaves himself, then the barber must not shave himself and if the barber does not shave himself, then the barber must shave himself.
You might think this paradox an oddity but, using this simple idea, Bertrand Russell changed the course of mathematical history and it is the fundamental paradox used to show computers are Turing limited.
The Russell Paradox
In the late 19th century, mathematicians began to think about the nature of numbers.
What is a number?
It is certainly not an object we can hold.
I can’t hold a two, unless it’s the brass number plate, for my front door. And, in that case I am holding one number plate, so I am not holding the idea of two, but rather the idea of one: one brass plate in the shape of a two.
The ‘idea’ of a number is to say something about the things I have in my hand: two apples, two oranges and two brass number plates. These are all sets of two things and ‘two’ is the collection of all these sets.
In 1890, Gottlob Frege completed his theory of sets. The project had taken him five years. Unfortunately, just before sending the book to the publisher, Bertrand Russell wrote to him and pointed out the following paradox. What about the set of sets that does not contain itself? Think about it...
In the late 19th century, mathematicians began to think about the nature of numbers.
What is a number?
It is certainly not an object we can hold.
I can’t hold a two, unless it’s the brass number plate, for my front door. And, in that case I am holding one number plate, so I am not holding the idea of two, but rather the idea of one: one brass plate in the shape of a two.
The ‘idea’ of a number is to say something about the things I have in my hand: two apples, two oranges and two brass number plates. These are all sets of two things and ‘two’ is the collection of all these sets.
In 1890, Gottlob Frege completed his theory of sets. The project had taken him five years. Unfortunately, just before sending the book to the publisher, Bertrand Russell wrote to him and pointed out the following paradox. What about the set of sets that does not contain itself? Think about it...
HOUSE_OVERSIGHT_015845
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