Past, present and future of set theory
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
What one can say about the past, present and future of set theory depends on what one expects or at least hopes set theory will accomplish. In order to gauge the early expectations, I begin with a quote from the inaugural lecture in 1903 of my mathematical grandfather, the internationally known Finnish mathematician Ernst Lindelöf. The subject of his lecture was – guess what – Cantor’s set theory. In his conclusion, Lindelöf says of Cantor’s results: For mathematics they have lent new tools and opened up new fields of research, they have thrown entirely new light on the foundations of analysis and brought clarity and order where there was only disorder and contradictions. Thus they have greatly contributed to the harmony that is the essence of mathematics, a harmony a grasp of which is the reward of mathematical research. (Quoted in Olli Lehto, Tieteen aatelia, Otava, Helsinki, 2008, p. 263) We can all agree with the compliments Lindelöf pays to set theory as an impressive specimen of mathematical research, including the theory of infinite cardinals and ordinals. But as far as the foundational role of set theory is concerned, in the perspective of the subsequent century his words read as an example of supreme historical irony. Far from bringing harmony into the foundations of mathematics, problems arising from set theory led to a schism between different schools of thought. Few mathematicians think of set theory as a tool for reaching new results outside set theory.. On the contrary, an interesting rich tradition called reverse mathematics takes significant mathematical results and asks what set-theoretical assumptions are needed to prove them. Set-theoretical paradoxes have greatly increased mathematicians’ concerns about contradictions instead of assuaging them. Many foundationalists would blandly deny that we have even now, more than a hundred years later, reached “clarity and order” about the foundations of analysis. What Lindelöf took to be the results of set theory thus were in reality so many hopes that set theory was expected to fulfill..
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library||
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Mujdat Pakkan & Varol Akman (1995). Issues in Commonsense Set Theory. Philosophical Explorations.
Gregory H. Moore (1980). Beyond First-Order Logic: The Historical Interplay Between Mathematical Logic and Axiomatic Set Theory. History and Philosophy of Logic 1 (1-2):95-137.
F. A. Muller (2001). Sets, Classes, and Categories. British Journal for the Philosophy of Science 52 (3):539-573.
Patrick Dehornoy (1996). Another Use of Set Theory. Bulletin of Symbolic Logic 2 (4):379-391.
John Mayberry (1994). What is Required of a Foundation for Mathematics? Philosophia Mathematica 2 (1):16-35.
P. T. Johnstone (1987). Notes on Logic and Set Theory. Cambridge University Press.
Peter Schreiber (1996). Mengenlehre—Vom Himmel Cantors Zur Theoria Prima Inter Pares. NTM International Journal of History and Ethics of Natural Sciences, Technology and Medicine 4 (1):129-143.
Stewart Shapiro (2000). Set-Theoretic Foundations. The Proceedings of the Twentieth World Congress of Philosophy 2000:183-196.
Makmiller Pedroso (2009). On Three Arguments Against Categorical Structuralism. Synthese 170 (1):21 - 31.
Added to index2009-09-11
Total downloads18 ( #92,698 of 1,101,139 )
Recent downloads (6 months)0
How can I increase my downloads?