£66.95

Borel Liftings of Borel Sets - Some Decidable and Undecidable Statements (Memoirs of the American Mathematical Society)

Geometry & Topology

Price data last checked 17 day(s) ago - will refresh soon

View at Amazon

We'll watch every seller, every day. One email when your price arrives.

It has never been this cheap. We have no record of a lower price.

£67 today · cheaper than every other day in the last 3 months

NEW HERE?

Amazon shows you one price. We show you all of them.

Tosheroon watches Amazon prices so you don't have to. Every product on Amazon has a price history — we make it visible. Set the price you'd actually pay, and we'll email you the second it gets there. No app, no account, one email.

WHAT'S ON THIS PAGE

↓ Price chart
when this has been cheap or pricey

↓ Forecast
where the price is heading next

↓ Statistics
all-time high & low, recent range

↑ Price alert
name your number, we'll email you

Price History & Forecast

Grey patches = out of stock. Cheaper = lower on the chart. Hover for exact prices.

Last 74 days · 74 data points (no recent data)

Historical

Generating forecast…

Price Distribution

Price distribution over 74 days • 1 price levels

Days at Price

Price Analysis

Most common price: £67 (74 days, 100.0%)

Price range: £67 - £67

Price levels: 1 different prices over 74 days

Description

One of the aims of this work is to investigate some natural properties of Borel sets which are undecidable in $ZFC$. The authors' starting point is the following elementary, though non-trivial result: Consider $X \subset 2omega\times2omega$, set $Y=\pi(X)$, where $\pi$ denotes the canonical projection of $2omega\times2omega$ onto the first factor, and suppose that $(\star)$ : ""Any compact subset of $Y$ is the projection of some compact subset of $X$"". If moreover $X$ is $\mathbf{\Pi 0 2$ then $(\star\star)$: ""The restriction of $\pi$ to some relatively closed subset of $X$ is perfect onto $Y$"" it follows that in the present case $Y$ is also $\mathbf{\Pi 0 2$. Notice that the reverse implication $(\star\star)\Rightarrow(\star)$ holds trivially for any $X$ and $Y$. But the implication $(\star)\Rightarrow (\star\star)$ for an arbitrary Borel set $X \subset 2omega\times2omega$ is equivalent to the statement ""$\forall \alpha\in \omegaomega, \,\aleph 1$ is inaccessible in $L(\alpha)$"". More precisely The authors prove that the validity of $(\star)\Rightarrow(\star\star)$ for all $X \in \varSigma0 {1 \xi 1 $, is equivalent to ""$\aleph \xi \aleph 1$"". However we shall show independently, that when $X$ is Borel one can, in $ZFC$, derive from $(\star)$ the weaker conclusion that $Y$ is also Borel and of the same Baire class as $X$. This last result solves an old problem about compact covering mappings. In fact these results are closely related to the following general boundedness principle Lift$(X, Y)$: ""If any compact subset of $Y$ admits a continuous lifting in $X$, then $Y$ admits a continuous lifting in $X$"", where by a lifting of $Z\subset \pi(X)$ in $X$ we mean a mapping on $Z$ whose graph is contained in $X$. The main result of this work will give the exact set theoretical strength of this principle depending on the descriptive complexity of $X$ and $Y$. The authors also prove a similar result for a variation of Lift$(X, Y)$ in which ""continuous liftings"" are replaced by ""Borel liftings"", and which answers a question of H. Friedman. Among other applications the authors obtain a complete solution to a problem which goes back to Lusin concerning the existence of $\mathbf{\Pi 1 1$ sets with all constituents in some given class $\mathbf{\Gamma $ of Borel sets, improving earlier results by J. Stern and R. Sami. The proof of the main result will rely on a nontrivial representation of Borel sets (in $ZFC$) of a new type, involving a large amount of ""abstract algebra"". This representation was initially developed for the purposes of this proof, but has several other applications.

Product Specifications

Format: paperback
ASIN: 0821839713
Category: Books > Subjects > Science, Nature & Maths > Mathematics > Geometry & Topology
Domain: Amazon UK
Release Date: 17 May 2007
Listed Since: 22 March 2007

Barcode

No barcode data available

Borel Liftings of Borel Sets - Some Decidable and Undecidable Statements (Memoirs of the American Mathematical Society)

It has never been this cheap. We have no record of a lower price.

Price History & Forecast

Price Distribution

Price Analysis

Description

Product Specifications

Barcode

Similar Products You Might Like

The Theory of Countable Borel Equivalence Relations: 234 (Cambridge Tracts in Mathematics, Series Number 234)

Ultrafilters Across Mathematics (Contemporary Mathematics): International Congress Ultramath 2008: Applications of Ultrafilters and Ultraproducts in Mathematics, June 1-7, 2008, Pisa, Italy

A Course on Borel Sets: 180 (Graduate Texts in Mathematics, 180)

Centenary of the Borel Conjecture (Contemporary Mathematics)

Selected Topics of Invariant Measures in Polish Groups (Mathematics Research Developments)

Descriptive Set Theory and Forcing: How to Prove Theorems about Borel Sets the Hard Way: 4 (Lecture Notes in Logic, Series Number 4)

Descriptive Set Theory and Forcing: How to prove theorems about Borel sets the hard way: 4 (Lecture Notes in Logic, 4)

Geometry of Subanalytic and Semialgebraic Sets: 150 (Progress in Mathematics, 150)

The Forcing Method in Set Theory: An Introduction via Boolean Valued Logic: 168 (UNITEXT, 168)

Dimension Theory: A Selection of Theorems and Counterexamples: 7 (Atlantis Studies in Mathematics, 7)

Structure and Randomness in Computability and Set Theory

Knowledge Representation Techniques: A Rough Set Approach: 202 (Studies in Fuzziness and Soft Computing, 202)

Many Valued Topology and its Applications

Prokhorov and Contemporary Probability Theory: In Honor of Yuri V. Prokhorov: 33 (Springer Proceedings in Mathematics & Statistics, 33)

Geometric Set Theory (Mathematical Surveys and Monographs)

Nonlinear Mathematics for Uncertainty and its Applications: 100 (Advances in Intelligent and Soft Computing, 100)

Nonlinear Mathematics for Uncertainty and its Applications: 100 (Advances in Intelligent and Soft Computing, 100)

Logic and Algebra: Logic and Algebra: 302 (Contemporary Mathematics)

A Bridge to Higher Mathematics: 43 (Textbooks in Mathematics)

Topics in Measure Theory and Real Analysis: 2 (Atlantis Studies in Mathematics)

Oeuvres Scientifiques - Collected Papers I: 1926-1951 (Springer Collected Works in Mathematics)

Introduction aux méthodes de Baire

Algebraic Formalization of Smart Systems: Theory and Practice: 91 (Smart Innovation, Systems and Technologies, 91)

The Logic of Sortals: A Conceptualist Approach: 408 (Synthese Library, 408)