S4F: The End of Infinity (Infinity of Fun Part III)
by David Siegel Bernstein, PhD
To conclude my series on infinity, I will now attempt to answer the question: does size does matter? Answer: maybe (at least for infinity . . . which is what I am talking about).
According to the mathematician Georg Cantor there is an infinite number of infinities including “little bits” of infinity. Meaning, not all infinities are created equal. No fair! Or is it? Who knows? Remember infinity is an idea not a number.
Consider the — artfully designed — Venn diagram below. It represents infinite sized sets nestled inside other infinite sets. The inner most set represents the universe of whole (counting) numbers, but as shown this is only a subset of integers (whole + negative numbers) which is a subset of rational numbers (whole + negative + fractions/decimals), which is a subset of real numbers (all the above + irrational numbers).
Cantor demonstrated that these sets cannot be completely matched up; what he called a lack of 1-1 correspondence. Simply put, if you were to keep using a whole number to cancel out an integer number, once you’ve exhausted the your stash of whole numbers you would still have integers left over after — even though you had an infinite number of whole numbers!
By the way, there are sets that lay beyond real numbers called transfinite numbers. Good luck thinking about those! If anyone is interested in learning more about them, let me knew and I’ll cover it in a future post.
See you next month in time and space with a new topic.
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Authors: David Siegel Bernstein. Form: Column. Length: 500 words. Editor who accepted this story: Previous Editors.
Author Bio
Abandoned Towers Content: S4F: Black Holes (Don’t Really) Suck S4F: Black Holes Suck S4F: Can Anyone Intelligent Communicate with Me? Other Than My Earthling Friends. Part II S4F: Can Anyone Intelligent Communicate with Me? Other Than My Earthling Friends. S4F: Do You Know the Time? S4F: Do You Know the Time? Part II S4F: Do You Know the Time? Part III S4F: Fiction by the Numbers S4F: From Higgs to Holograms S4F: I Have Virtually Nothing to Say About Zero-Point Energy S4F: If You are Uncertain—Call a Quantum Mechanic for a Fix S4F: Infinity for Fun S4F: Infinity of Fun, Part II S4F: Invisibility, Science not Magic S4F: Meet David Siegel Bernstein – Forensic Statistician S4F: More or Less Human S4F: Much Ado About Nothing This Month S4F: Now Playing at the Opera—The Faster than Light Particle Experiment S4F: Paradox Found S4F: Paradox Lost S4F: Parallel Worlds S4F: Splicing Science from Fiction S4F: The End of Infinity (Infinity of Fun Part III) S4F: We Hold Universal Truths to Be Not Self-Evident S4F: What’s the Matter with Antimatter?
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March 8th, 2012 at 6:43 pm
Interesting article!
Stealth is easier than cloaking, and we already have the technology.
March 30th, 2012 at 7:41 am
William you are absolutely correct. Invisibility (a descendant of Stealth) is not only easier to achieve, but a future topic. I thought I’d go over invisibility first to please those who enjoy Harry Potter, but want to write hard science fiction.
April 5th, 2012 at 5:39 am
Fascinating post. We are so much empty space and mostly water. Amazing to think how we can be beings of intelligence and creativity.