Mosterin, Jesus
(2004)
How Set Theory Impinges on Logic.
UNSPECIFIED.
Abstract
Standard (classical) logic is not independent of set theory. Which formulas are valid in logic depends on which sets we assume to exist in our settheoretical universe. Secondorder logic is just set theory in disguise. The typically logical notions of validity and consequence are not well defined in secondorder logic, at least as long as there are open issues in set theory. Such contentious issues in set theory as the axiom of choice, the continuum hypothesis or the existence of inaccessible cardinals, can be equivalently transformed into question about the logical validity of pure sentences of secondorder logic, where “pure” means that they only contain logical symbols and bound variables. Even standard firstorder logic depends on the acceptance on infinite sets in our settheoretical universe. Should we choose to admit only finite sets, the number of logically valid pure firstorder formulas would increase dramatically and firstorder logic would not be recursively enumerable any longer.
Item Type: 
Other

Creators: 
Creators  Email  ORCID 

Mosterin, Jesus   

Additional Information: 
Published in Paul Weingartner (ed.), Alternative Logics: Do Sciences Need Them? BerlinHeidelbergNew York, 2004, pp. 5563. 
Keywords: 
Models, settheoretical universe, infinite, firstorder logic, secondorder logic, set theory, continuum hypothesis 
Subjects: 
General Issues > Models and Idealization Specific Sciences > Mathematics 
Depositing User: 
Jesus Mosterin

Date Deposited: 
16 Feb 2004 
Last Modified: 
07 Oct 2010 15:12 
Item ID: 
1620 
Public Domain: 
No 
URI: 
http://philsciarchive.pitt.edu/id/eprint/1620 
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