November 2025
Beginner
416 pages
8h 6m
English
Content preview from Logic For Dummies, 2nd Edition
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Chapter 22
Paradox and Axiomatic Systems
IN THIS CHAPTER
Grasping the basics of set theory and Russell’s paradox
Seeing how the SL axiomatic system measures up
Appreciating consistency and completeness
Limiting mathematics with Gödel’s incompleteness theorem
Consider the following statement:
This statement is false.
If the statement is true, then it must be false. However, if it’s false, it must be true. This problem, called the liar paradox, dates back to the Greeks.
At first glance, the liar paradox seems no more than a curiosity. But paradoxes like this one have surfaced and resurfaced in various forms, causing trouble for logicians and challenging them to search for ways to resolve the trouble.
In this chapter, I explain Russell’s paradox (a modification of the liar paradox), which forced logicians to undertake a radical restructuring of the foundations of set theory and logic. This paradox leads to a discussion of Principia Mathematica, which is an attempt to formulate set theory, logic, and ultimately all of math based upon a set of assumptions called axioms. You also see ...
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I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.Amir M.
I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.