The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Higher infinity and the foundations of mathematics Joel David Hamkins The City University of New York College of Staten Island & The CUNY Graduate Center Mathematics, Philosophy, Computer Science American Association for the Advancement of Science Pacific Division 95th annual meeting Riverside, CA, June 18-20, 2014 Higher infinity and foundations of math Joel David Hamkins, New York The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Higher infinity and the foundations of mathematics The story of infinity and what is going on in the foundations of mathematics Different kinds of infinity, reaching up to the higher infinite Pervasive independence phenomenon, meaning numerous fundamental questions in set theory and mathematics are neither provable nor refutable Troubling philosophical questions at the core of mathematics concerning the nature and meaning of truth Higher infinity and foundations of math Joel David Hamkins, New York 0 1 10 100 million 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000 The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Large finite numbers Large finite numbers as a metaphor for infinity. What is the largest number you know? Higher infinity and foundations of math Joel David Hamkins, New York 1 10 100 million 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000 The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Large finite numbers Large finite numbers as a metaphor for infinity. What is the largest number you know? 0 Higher infinity and foundations of math Joel David Hamkins, New York 10 100 million 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000 The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Large finite numbers Large finite numbers as a metaphor for infinity. What is the largest number you know? 0 1 Higher infinity and foundations of math Joel David Hamkins, New York 100 million 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000 The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Large finite numbers Large finite numbers as a metaphor for infinity. What is the largest number you know? 0 1 10 Higher infinity and foundations of math Joel David Hamkins, New York million 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000 The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Large finite numbers Large finite numbers as a metaphor for infinity. What is the largest number you know? 0 1 10 100 Higher infinity and foundations of math Joel David Hamkins, New York 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000 The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Large finite numbers Large finite numbers as a metaphor for infinity. What is the largest number you know? 0 1 10 100 million Higher infinity and foundations of math Joel David Hamkins, New York 100 zeros z }| { = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000 The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Large finite numbers Large finite numbers as a metaphor for infinity. What is the largest number you know? 0 1 10 100 million googol Higher infinity and foundations of math Joel David Hamkins, New York googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000 The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Large finite numbers Large finite numbers as a metaphor for infinity. What is the largest number you know? 0 1 10 100 million 100 zeros z }| { googol = 10100 = 100000 ··· 00 Higher infinity and foundations of math Joel David Hamkins, New York googol zeros 100 z }| { = 10googol = 1010 = 1000000000 ······ 0000 The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Large finite numbers Large finite numbers as a metaphor for infinity. What is the largest number you know? 0 1 10 100 million 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol plex Higher infinity and foundations of math Joel David Hamkins, New York The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Large finite numbers Large finite numbers as a metaphor for infinity. What is the largest number you know? 0 1 10 100 million 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000 Higher infinity and foundations of math Joel David Hamkins, New York googol bang = (googol)! = 1 · 2 · 3 ··· googol Which is larger? googol bang plex = 10(googol)! googol plex bang = 10googol! Which is larger? googol bang plex bang plex plex googol bang bang plex bang bang The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Getting larger googol plex = 10googol Higher infinity and foundations of math Joel David Hamkins, New York = (googol)! = 1 · 2 · 3 ··· googol Which is larger? googol bang plex = 10(googol)! googol plex bang = 10googol! Which is larger? googol bang plex bang plex plex googol bang bang plex bang bang The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Getting larger googol plex = 10googol googol bang Higher infinity and foundations of math Joel David Hamkins, New York googol bang plex = 10(googol)! googol plex bang = 10googol! Which is larger? googol bang plex bang plex plex googol bang bang plex bang bang The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Getting larger googol plex = 10googol googol bang = (googol)! = 1 · 2 · 3 ··· googol Which is larger? Higher infinity and foundations of math Joel David Hamkins, New York = 10(googol)! googol plex bang = 10googol! Which is larger? googol bang plex bang plex plex googol bang bang plex bang bang The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Getting larger googol plex = 10googol googol bang = (googol)! = 1 · 2 · 3 ··· googol Which is larger? googol bang plex Higher infinity and foundations of math Joel David Hamkins, New York googol plex bang = 10googol! Which is larger? googol bang plex bang plex plex googol bang bang plex bang bang The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Getting larger googol plex = 10googol googol bang = (googol)! = 1 · 2 · 3 ··· googol Which is larger? googol bang plex = 10(googol)! Higher infinity and foundations of math Joel David Hamkins, New York = 10googol! Which is larger? googol bang plex bang plex plex googol bang bang plex bang bang The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Getting larger googol plex = 10googol googol bang = (googol)! = 1 · 2 · 3 ··· googol Which is larger? googol bang plex = 10(googol)! googol plex bang Higher infinity and foundations of math Joel David Hamkins, New York googol bang plex bang plex plex googol bang bang plex bang bang The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Getting larger googol plex = 10googol googol bang = (googol)! = 1 · 2 · 3 ··· googol Which is larger? googol bang plex = 10(googol)! googol plex bang = 10googol! Which is larger? Higher infinity and foundations of math Joel David Hamkins, New York The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Getting larger googol plex = 10googol googol bang = (googol)! = 1 · 2 · 3 ··· googol Which is larger? googol bang plex = 10(googol)! googol plex bang = 10googol! Which is larger? googol bang plex bang plex plex googol bang bang plex bang bang Higher infinity and foundations of math Joel David Hamkins, New York 10 ·· 10· g googol = 1010 Consider all expressions in the googol plex bang stack language. googol stack bang plex plex bang googol plex stack bang plex bang googol bang stack plex Question Is there a feasible computable procedure to say which of any two such expressions describes the larger number? This is an open mathematical question. However, we do know how to compare expressions having at most a googol terms. The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Truly vast numbers googol stack Higher infinity and foundations of math Joel David Hamkins, New York Consider all expressions in the googol plex bang stack language. googol stack bang plex plex bang googol plex stack bang plex bang googol bang stack plex Question Is there a feasible computable procedure to say which of any two such expressions describes the larger number? This is an open mathematical question. However, we do know how to compare expressions having at most a googol terms. The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Truly vast numbers 10 ·· 10· g googol googol stack = 1010 Higher infinity and foundations of math Joel David Hamkins, New York googol stack bang plex plex bang googol plex stack bang plex bang googol bang stack plex Question Is there a feasible computable procedure to say which of any two such expressions describes the larger number? This is an open mathematical question. However, we do know how to compare expressions having at most a googol terms. The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence Truly vast numbers 10 ·· 10· g googol googol stack = 1010 Consider all expressions in the googol plex bang stack language. Higher infinity and foundations of math Joel David Hamkins, New York Question Is there a feasible computable procedure to say which of any two such expressions describes the larger number? This is an open mathematical question.