# The Infinite

By A. W. Moore | Go to book overview

CHAPTER 12

Gödel’s Theorem

The human mind is incapable of formulating…all its mathematical intuitions, ie., if it has succeeded in formulating some of them, this very fact yields new intuitive knowledge, eg., the consistency of this formalism. This fact may be called the ‘incompletability’ of mathematics.

(Kurt Gödel)

Gödel’s theorem is one of the most profound results in pure mathematics. When it was first published, in 1931, it had a devastating impact. On the one hand, it laid waste a variety of firmly held convictions and initiated a struggle that has been going on ever since to come to terms with its mathematical and philosophical implications. On the other hand, it took the breath away for its sheer beauty. My aim in this chapter is to present an outline of the theorem and to say what some of its implications are for our own enquiry.

In a nutshell, it concerns the Euclidean paradigm—the paradigm of axiomatization. It is possible, we know, to devise a finite stock of fundamental principles or axioms from which all of the infinitely many truths of Greek geometry can be derived: this is the Euclidean paradigm. 1 Prior to 1931 many people had assumed that what was possible in geometry must be possible anywhere else in mathematics (and perhaps in non-mathematical contexts too); the paradigm must represent the very essence of mathematical method. 2 One of the reasons for this relates back to our discussion in the last chapter. Suppose we grant that the meaning of a mathematical expression has to be grasped in terms of how it figures in the truths of a formal theory. Then must there not be some way of ‘capturing’ these truths and providing them with a finite characterization—precisely what an axiomatization (and that alone?) can supply? How else could anyone assimilate the truths and grasp the expression’s meaning? Again, relatedly, do we have any sense of mathematical truth apart from mathematical provability? When we say that a given mathematical state-

-172-

If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

#### Items saved from this book

This book has been saved
Highlights (0)
Some of your highlights are legacy items.
Citations (0)
Some of your citations are legacy items.
Notes (0)
Bookmarks (0)

You have no saved items from this book

Project items include:
• Saved book/article
• Highlights
• Quotes/citations
• Notes
• Bookmarks
Notes

#### Cited page

Style
Citations are available only to our active members.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

#### Cited page

The Infinite

• Title Page iii
• Contents vii
• Preface to the Second Edition xi
• Preface xx
• Introduction: Paradoxes of the Infinite 1
• Part One - The History 15
• Chapter 1 - Early Greek Thought 17
• Chapter 2 - Aristotle 34
• Chapter 3 - Medieval and Renaissance Thought 45
• Chapter 4 - The Calculus 57
• Chapter 5 - The Rationalists and the Empiricists 75
• Chapter 6 - Kant 84
• Chapter 7 - Post-Kantian Metaphysics of the Infinite 96
• Chapter 8 - The Mathematics of the Infinite, and the Impact of Cantor 110
• Chapter 9 - Reactions 131
• Part Two - Infinity Assessed 145
• Chapter 10 - Transfinite Mathematics 147
• Chapter 11 - The Löwenheim-Skolem Theorem 159
• Chapter 12 - Gödel's Theorem 172
• Chapter 13 - Saying and Showing 186
• Chapter 14 - Infinity Assessed. the History Reassessed 201
• Chapter 15 - Human Finitude 218
• Glossary 234
• Bibliography 250
• Index 261
Settings

#### Settings

Typeface
Text size Reset View mode
Search within

Look up

#### Look up a word

• Dictionary
• Thesaurus
Please submit a word or phrase above.

Why can't I print more than one page at a time?

Help
Full screen
/ 268

### How to highlight and cite specific passages

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

## Cited passage

Style
Citations are available only to our active members.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

## Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.