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edited Apr 30, 2010 at 2:45

François G. Dorais

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The use of forcing in Set Theory is to investigate the Zermelo-Fraenkel axioms and their consequences. This is a perfectly valid use of Model Theory — the Completeness Theorem says that a statement φ is a consequence of ZFC if and only if φ is true in every model of ZFC. If one can produce a forcing poset that forces φ to be false, then we know that if ZFC is consistent then φ is not a consequence of ZFC since any suitable model can be extended to a model of ZFC in which φ is false. If another forcing forces φ to be true, then we know that φ is independent of ZFC.

I'm gathering from your question that you're a Platonist (I'm agnostic but I'll play along). This demonstration of independence via forcing says little about the truth of φ in the Platonic Universe. It only says that further information than the axioms of ZFC is needed to determine the truth of φ. However, these investigations over the past half-century have led to some insight on what statements are indeed true in the Platonic Universe. For example, see these two articles by Woodin (AMS Notices, 2001) where he discusses the status of the Continuum Hypothesis.

The use of forcing in Set Theory is to investigate the Zermelo-Fraenkel axioms and their consequences. This is a perfectly valid use of Model Theory — the Completeness Theorem says that a statement φ is a consequence of ZFC if and only if φ is true in every model of ZFC. If one can produce a forcing poset that forces φ to be false, then we know that if ZFC is consistent then φ is not a consequence of ZFC since any suitable model can be extended to a model of ZFC in which φ is false. If another forcing forces φ to be true, then we know that φ is independent of ZFC.

I'm gathering from your question that you're a Platonist (I'm agnostic). This demonstration of independence via forcing says little about the truth of φ in the Platonic Universe. It only says that further information than the axioms of ZFC is needed to determine the truth of φ. However, these investigations over the past half-century have led to some insight on what statements are indeed true in the Platonic Universe. For example, see these two articles by Woodin (AMS Notices, 2001) where he discusses the status of the Continuum Hypothesis.

The use of forcing in Set Theory is to investigate the Zermelo-Fraenkel axioms and their consequences. This is a perfectly valid use of Model Theory — the Completeness Theorem says that a statement φ is a consequence of ZFC if and only if φ is true in every model of ZFC. If one can produce a forcing poset that forces φ to be false, then we know that if ZFC is consistent then φ is not a consequence of ZFC since any suitable model can be extended to a model of ZFC in which φ is false. If another forcing forces φ to be true, then we know that φ is independent of ZFC.

I'm gathering from your question that you're a Platonist (I'm agnostic but I'll play along). This demonstration of independence via forcing says little about the truth of φ in the Platonic Universe. It only says that further information than the axioms of ZFC is needed to determine the truth of φ. However, these investigations over the past half-century have led to some insight on what statements are indeed true in the Platonic Universe. For example, see these two articles by Woodin (AMS Notices, 2001) where he discusses the status of the Continuum Hypothesis.

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created Apr 30, 2010 at 2:37

François G. Dorais

44.4k
6
150
233

The use of forcing in Set Theory is to investigate the Zermelo-Fraenkel axioms and their consequences. This is a perfectly valid use of Model Theory — the Completeness Theorem says that a statement φ is a consequence of ZFC if and only if φ is true in every model of ZFC. If one can produce a forcing poset that forces φ to be false, then we know that if ZFC is consistent then φ is not a consequence of ZFC since any suitable model can be extended to a model of ZFC in which φ is false. If another forcing forces φ to be true, then we know that φ is independent of ZFC.

I'm gathering from your question that you're a Platonist (I'm agnostic). This demonstration of independence via forcing says little about the truth of φ in the Platonic Universe. It only says that further information than the axioms of ZFC is needed to determine the truth of φ. However, these investigations over the past half-century have led to some insight on what statements are indeed true in the Platonic Universe. For example, see these two articles by Woodin (AMS Notices, 2001) where he discusses the status of the Continuum Hypothesis.