18194 | 'Forcing' can produce new models of ZFC from old models |

18195 | A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy |

13011 | New axioms are being sought, to determine the size of the continuum |

13014 | Extensional sets are clearer, simpler, unique and expressive |

13013 | The Axiom of Extensionality seems to be analytic |

13022 | Infinite sets are essential for giving an account of the real numbers |

13021 | The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics |

18191 | Axiom of Infinity: completed infinite collections can be treated mathematically |

13023 | The Power Set Axiom is needed for, and supported by, accounts of the continuum |

13028 | Replacement was added when some advanced theorems seemed to need it |

18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy |

13024 | Efforts to prove the Axiom of Choice have failed |

13026 | A large array of theorems depend on the Axiom of Choice |

13025 | Modern views say the Choice set exists, even if it can't be constructed |

17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres |

18169 | Axiom of Reducibility: propositional functions are extensionally predicative |

13019 | The Iterative Conception says everything appears at a stage, derived from the preceding appearances |

13018 | Limitation of Size is a vague intuition that over-large sets may generate paradoxes |

17824 | The master science is physical objects divided into sets |

8755 | Maddy replaces pure sets with just objects and perceived sets of objects |

10594 | Henkin semantics is more plausible for plural logic than for second-order logic |

17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying |

18168 | 'Propositional functions' are propositions with a variable as subject or predicate |

17625 | If two mathematical themes coincide, that suggest a single deep truth |

17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization |

18171 | Cantor and Dedekind brought completed infinities into mathematics |

18190 | Completed infinities resulted from giving foundations to calculus |

17615 | Every infinite set of reals is either countable or of the same size as the full set of reals |

18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities |

18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size |

18175 | For any cardinal there is always a larger one (so there is no set of all sets) |

18196 | An 'inaccessible' cardinal cannot be reached by union sets or power sets |

18172 | Infinity has degrees, and large cardinals are the heart of set theory |

18187 | Theorems about limits could only be proved once the real numbers were understood |

18182 | The extension of concepts is not important to me |

18177 | In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets |

18164 | Frege solves the Caesar problem by explicitly defining each number |

18178 | For Zermelo the successor of n is {n} (rather than n U {n}) |

18179 | For Von Neumann the successor of n is n U {n} (rather than {n}) |

18180 | Von Neumann numbers are preferred, because they continue into the transfinite |

17825 | Set theory (unlike the Peano postulates) can explain why multiplication is commutative |

17826 | Standardly, numbers are said to be sets, which is neat ontology and epistemology |

17828 | Numbers are properties of sets, just as lengths are properties of physical objects |

18183 | Set theory brings mathematics into one arena, where interrelations become clearer |

18188 | The line of rationals has gaps, but set theory provided an ordered continuum |

17618 | Set-theory tracks the contours of mathematical depth and fruitfulness |

10718 | A natural number is a property of sets |

18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms |

18186 | Identifying geometric points with real numbers revealed the power of set theory |

18185 | Unified set theory gives a final court of appeal for mathematics |

18184 | Making set theory foundational to mathematics leads to very fruitful axioms |

17827 | Sets exist where their elements are, but numbers are more like universals |

17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties |

17823 | If mathematical objects exist, how can we know them, and which objects are they? |

8756 | Intuition doesn't support much mathematics, and we should question its reliability |

17733 | We know mind-independent mathematical truths through sets, which rest on experience |

18204 | Scientists posit as few entities as possible, but set theorist posit as many as possible |

18207 | Maybe applications of continuum mathematics are all idealisations |

17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics |

17829 | Number words are unusual as adjectives; we don't say 'is five', and numbers always come first |

18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number |

18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real |

18206 | Science idealises the earth's surface, the oceans, continuities, and liquids |