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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Godel's Incompleteness Theorems</title>
<link rel="stylesheet" href="./style.css">
</head>
<body>
<nav>
<ol>
<li>
<a href="./index.html#sec1">A History and a Result</a>
<ol>
<li><a href="./index.html#sec1-1">The Axiomatic System is Developed</a></li>
<li><a href="./index.html#sec1-2">The Odd One Out</a></li>
<li><a href="./index.html#sec1-3">An Alternative Geometry</a></li>
<li><a href="./index.html#sec1-4">A Bigger Problem</a></li>
<li>
<a href="./index.html#sec1-5">A New Foundation for Mathematics</a>
<ol>
<li><a href="./index.html#sec1-5-1">Arithmetic, Set Theory, and Logic</a></li>
</ol>
</li>
<li><a href="./index.html#sec1-6">A Crisis in the Foundation</a></li>
<li><a href="./index.html#sec1-7">A New Theory of Sets</a></li>
<li><a href="./index.html#sec1-8">The Culmination of Mathematical History</a></li>
<li><a href="./index.html#sec1-9">The Result</a></li>
</ol>
</li>
<li>
<a href="./part2.html#sec2">The Proof</a>
<ol>
<li>
<a href="./part2.html#sec2-1">Necessary Considerations</a>
<ol>
<li><a href="./part2.html#sec2-1-1">Logic</a></li>
<li><a href="./part2.html#sec2-1-2">Primitive Recursion</a></li>
</ol>
</li>
<li>
<a href="./part2.html#sec2-2">The Proof</a>
<ol>
<li><a href="./part2.html#sec2-2-1">Defining the System P</a></li>
<li><a href="./part2.html#sec2-2-2">Gödel Numbering</a></li>
<li><a href="./part2.html#sec2-2-3">A Primitive Recursive Series</a></li>
<li><a href="./part2.html#sec2-2-4">Constructing the Undecidable</a></li>
<li><a href="./part2.html#sec2-2-5">A Problem With Consistency</a></li>
</ol>
</li>
</ol>
</li>
<li>
<a href="./part3.html#sec3">Closing Thoughts</a>
<ol>
<li><a href="./part3.html#sec3-1">Human Reasoning</a></li>
<li><a href="./part3.html#sec3-2">Philosophical Import</a></li>
<li><a href="./part3.html#sec3-3">What is Mathematics?</a></li>
</ol>
</li>
<li>
<a href="#sec4">Bibliography</a>
<ol>
<li><a href="#sec4-1">Cite this Paper</a></li>
</ol>
</li>
</ol>
</nav>
<header><a href="./part3.html">< Closing Thoughts</a></header>
<h2 id="sec4"><small>4</small> <a href="#sec4" class="section">Bibliography</a></h2>
<p>Kurt Gödel, <em>On Formally Undecidable Propositions of Principia Mathematica and Related Systems</em>, Basic Books, 1962.</p>
<p>Ernest Nagel, James R. Newman, <em>Gödel's Proof</em>, New York University Press, 2001.</p>
<p>Carl Boyer, <em>A History of Mathematics</em>, Princeton University Press, 1985.</p>
<p>Proclus, <em>A Commentary on the First Book of Euclid’s Elements</em>, Princeton University Press, 1992.</p>
<p>Alexander Bogomolny, <em>Attempts to Prove Euclid’s Fifth Postulate</em>, Interactive Mathematics Miscellany and Puzzles, 2018.</p>
<p>Peter Wolff, <em>Breakthroughs in Mathematics</em>, New American Library, New York, NY, 1970.</p>
<p>Morris Kline, <em>Mathematics and Western Culture</em>, Oxford University Press, 1953.</p>
<p>Georg Cantor, <em>Contributions in Support of a Transfinite Set Theory</em> Dover Publications, 1915.</p>
<p>Georg Cantor, <em>On a Property of the Collection of all Real Algebraic Numbers</em>, 1874.</p>
<p>Gottlob Frege, Michael Beaney, <em>The Frege Reader</em>, Wiley, 1997.</p>
<p>Bertrand Russell, Alfred North Whitehead, <em>Principia Mathematica</em>, Cambridge University Press, 1910.</p>
<p>Douglas Hofstadter, <em>Gödel, Escher, Bach</em>, Basic Books, 1979.</p>
<h3 id="sec4-1"><small>4.1</small> <a href="#sec4-1" class="section">Cite this Paper</a></h3>
<p>You can cite this paper as</p>
<p class="center">Jack Evoniuk, <em>Gödel's Incompleteness Theorems</em>, 2020.</p>
<p>It would likely be best to include a link as well, as the above citation isn't exactly very specific.</p>
</body>