green numbers give full details | back to texts | expand these ideas

18137 | Impredicative definitions are wrong, because they change the set that is being defined? |

18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism |

18114 | There is no single agreed structure for set theory |

18107 | A 'proper class' cannot be a member of anything |

18115 | We could add axioms to make sets either as small or as large as possible |

18139 | The Axiom of Choice relies on reference to sets that we are unable to describe |

18105 | Replacement enforces a 'limitation of size' test for the existence of sets |

18108 | First-order logic is not decidable: there is no test of whether any formula is valid |

18109 | The completeness of first-order logic implies its compactness |

18123 | Substitutional quantification is just standard if all objects in the domain have a name |

18120 | The Deduction Theorem is what licenses a system of natural deduction |

18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' |

18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 |

18100 | ω + 1 is a new ordinal, but its cardinality is unchanged |

18102 | A cardinal is the earliest ordinal that has that number of predecessors |

18106 | Aleph-1 is the first ordinal that exceeds aleph-0 |

18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms |

18099 | The number of reals is the number of subsets of the natural numbers |

18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number |

18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers |

18156 | Modern axioms of geometry do not need the real numbers |

18097 | The Peano Axioms describe a unique structure |

18148 | Hume's Principle is a definition with existential claims, and won't explain numbers |

18149 | There are many criteria for the identity of numbers |

18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it |

18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! |

18116 | Numbers can't be positions, if nothing decides what position a given number has |

18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties |

18141 | Nominalism about mathematics is either reductionist, or fictionalist |

18157 | Nominalism as based on application of numbers is no good, because there are too many applications |

18150 | Actual measurement could never require the precision of the real numbers |

18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... |

18127 | Simple type theory has 'levels', but ramified type theory has 'orders' |

18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number |

18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job |

18129 | Many crucial logicist definitions are in fact impredicative |

18146 | If Hume's Principle is the whole story, that implies structuralism |

18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality |

18159 | Higher cardinalities in sets are just fairy stories |

18155 | A fairy tale may give predictions, but only a true theory can give explanations |

18140 | The best version of conceptualism is predicativism |

18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values |

18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them |

18132 | The predicativity restriction makes a difference with the real numbers |

18133 | The usual definitions of identity and of natural numbers are impredicative |

18134 | Predicativism makes theories of huge cardinals impossible |

18136 | If we can only think of what we can describe, predicativism may be implied |

18135 | If mathematics rests on science, predicativism may be the best approach |

18121 | In logic a proposition means the same when it is and when it is not asserted |