Generic saturation
Assuming that ORD is ω + ω-Erdös we show that if a class forcing amenable to L (an L-forcing) has a generic then it has one definable in a set-generic extension of L [ O # ]. In fact we may choose such a generic to be periodic in the sense that it preserve the indiscernibility of a final segment of...
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| Veröffentlicht in: | The Journal of symbolic logic 1998-03, Vol.63 (1), p.158-162 |
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| container_title | The Journal of symbolic logic |
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| creator | Friedman, Sy D. |
| description | Assuming that ORD is ω + ω-Erdös we show that if a class forcing amenable to
L
(an L-forcing) has a generic then it has one definable in a set-generic extension of
L
[
O
#
]. In fact we may choose such a generic to be
periodic
in the sense that it preserve the indiscernibility of a final segment of a periodic subclass of the Silver indiscernibles, and therefore to be
almost codable
in the sense that it is definable from a real which is generic for an
L
-forcing (and which belongs to a set-generic extension of
L
[
0
#
]). This result is best possible in the sense that for any countable ordinal α there is an
L
-forcing which has generics but none periodic of period ≤ α. However, we do not know if an assumption beyond ZFC+“
O
#
exists” is actually necessary for these results.
Let
P
denote a class forcing definable over an amenable ground model 〈
L
,
A
〉 and assume that
O
#
exists.
D
efinition
.
P
is
relevant
if
P
has a generic definable in
L
[
0
#
].
P
is
almost relevant
if
P
has a generic definable in a set-generic extension of
L
[
0
#
].
R
emark
. The reverse Easton product of Cohen forcings 2 |
| doi_str_mv | 10.2307/2586594 |
| format | Article |
| fullrecord | <record><control><sourceid>jstor_proje</sourceid><recordid>TN_cdi_projecteuclid_primary_oai_CULeuclid_euclid_jsl_1183745464</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><jstor_id>2586594</jstor_id><sourcerecordid>2586594</sourcerecordid><originalsourceid>FETCH-LOGICAL-c337t-a85ab0d33aaf5ab31d39f473b6474af0ff66f064e69705909772637af7deff9e3</originalsourceid><addsrcrecordid>eNp90E1LxDAQBuAgCq6rCP4CQcFTdZLJV2_qoquyIILrNWTbBFrXrSYp6L830rJHTzMMD-8MQ8gxhUuGoK6Y0FKUfIdMaMmxEFrLXTIBYKzgmrJ9chBjCwDZ6Ak5mbuNC011Gm3qg01Ntzkke96uozsa65Qs7-9eZw_F4nn-OLtZFBWiSoXVwq6gRrTW5w5pjaXnCleSK249eC-lB8mdLFVeBqVSTKKyXtXO-9LhlFwPuZ-ha12VXF-tm9p8hubDhh_T2cbMlotxOpY2rg2lGhUXXPIccbaN-OpdTKbt-rDJVxsqkVPGUGBWF4OqQhdjcH67g4L5-5kZf5bl-SDbmLrwDysG1sTkvrfMhncjFSph5PzF3MITZW9aGMBfLrh19w</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1634122353</pqid></control><display><type>article</type><title>Generic saturation</title><source>Jstor Complete Legacy</source><source>Periodicals Index Online</source><source>JSTOR Mathematics & Statistics</source><creator>Friedman, Sy D.</creator><creatorcontrib>Friedman, Sy D.</creatorcontrib><description>Assuming that ORD is ω + ω-Erdös we show that if a class forcing amenable to
L
(an L-forcing) has a generic then it has one definable in a set-generic extension of
L
[
O
#
]. In fact we may choose such a generic to be
periodic
in the sense that it preserve the indiscernibility of a final segment of a periodic subclass of the Silver indiscernibles, and therefore to be
almost codable
in the sense that it is definable from a real which is generic for an
L
-forcing (and which belongs to a set-generic extension of
L
[
0
#
]). This result is best possible in the sense that for any countable ordinal α there is an
L
-forcing which has generics but none periodic of period ≤ α. However, we do not know if an assumption beyond ZFC+“
O
#
exists” is actually necessary for these results.
Let
P
denote a class forcing definable over an amenable ground model 〈
L
,
A
〉 and assume that
O
#
exists.
D
efinition
.
P
is
relevant
if
P
has a generic definable in
L
[
0
#
].
P
is
almost relevant
if
P
has a generic definable in a set-generic extension of
L
[
0
#
].
R
emark
. The reverse Easton product of Cohen forcings 2
<κ
, κ regular is relevant. So are the Easton product and the full product, provided κ is restricted to the successor cardinals. See Chapter 3, Section Two of Friedman [3]. Of course any set-forcing (in
L
) is almost relevant.
Definition.
κ
is α-Erdös if whenever
C
is CUB in κ and
f
: [C]
<ω
→ κ is regressive (i.e.,
f
(
a
) < min(
a
)) then
f
has a homogeneous set of ordertype α.</description><identifier>ISSN: 0022-4812</identifier><identifier>EISSN: 1943-5886</identifier><identifier>DOI: 10.2307/2586594</identifier><language>eng</language><publisher>New York, USA: Cambridge University Press</publisher><subject>Cofinality ; Indiscernibles ; Mathematical theorems ; Periodicity</subject><ispartof>The Journal of symbolic logic, 1998-03, Vol.63 (1), p.158-162</ispartof><rights>Copyright 1998 Association for Symbolic Logic Inc</rights><rights>Copyright 1998 Association for Symbolic Logic</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c337t-a85ab0d33aaf5ab31d39f473b6474af0ff66f064e69705909772637af7deff9e3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/2586594$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/2586594$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>230,314,776,780,799,828,881,27848,27903,27904,57995,57999,58228,58232</link.rule.ids></links><search><creatorcontrib>Friedman, Sy D.</creatorcontrib><title>Generic saturation</title><title>The Journal of symbolic logic</title><description>Assuming that ORD is ω + ω-Erdös we show that if a class forcing amenable to
L
(an L-forcing) has a generic then it has one definable in a set-generic extension of
L
[
O
#
]. In fact we may choose such a generic to be
periodic
in the sense that it preserve the indiscernibility of a final segment of a periodic subclass of the Silver indiscernibles, and therefore to be
almost codable
in the sense that it is definable from a real which is generic for an
L
-forcing (and which belongs to a set-generic extension of
L
[
0
#
]). This result is best possible in the sense that for any countable ordinal α there is an
L
-forcing which has generics but none periodic of period ≤ α. However, we do not know if an assumption beyond ZFC+“
O
#
exists” is actually necessary for these results.
Let
P
denote a class forcing definable over an amenable ground model 〈
L
,
A
〉 and assume that
O
#
exists.
D
efinition
.
P
is
relevant
if
P
has a generic definable in
L
[
0
#
].
P
is
almost relevant
if
P
has a generic definable in a set-generic extension of
L
[
0
#
].
R
emark
. The reverse Easton product of Cohen forcings 2
<κ
, κ regular is relevant. So are the Easton product and the full product, provided κ is restricted to the successor cardinals. See Chapter 3, Section Two of Friedman [3]. Of course any set-forcing (in
L
) is almost relevant.
Definition.
κ
is α-Erdös if whenever
C
is CUB in κ and
f
: [C]
<ω
→ κ is regressive (i.e.,
f
(
a
) < min(
a
)) then
f
has a homogeneous set of ordertype α.</description><subject>Cofinality</subject><subject>Indiscernibles</subject><subject>Mathematical theorems</subject><subject>Periodicity</subject><issn>0022-4812</issn><issn>1943-5886</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1998</creationdate><recordtype>article</recordtype><sourceid>K30</sourceid><recordid>eNp90E1LxDAQBuAgCq6rCP4CQcFTdZLJV2_qoquyIILrNWTbBFrXrSYp6L830rJHTzMMD-8MQ8gxhUuGoK6Y0FKUfIdMaMmxEFrLXTIBYKzgmrJ9chBjCwDZ6Ak5mbuNC011Gm3qg01Ntzkke96uozsa65Qs7-9eZw_F4nn-OLtZFBWiSoXVwq6gRrTW5w5pjaXnCleSK249eC-lB8mdLFVeBqVSTKKyXtXO-9LhlFwPuZ-ha12VXF-tm9p8hubDhh_T2cbMlotxOpY2rg2lGhUXXPIccbaN-OpdTKbt-rDJVxsqkVPGUGBWF4OqQhdjcH67g4L5-5kZf5bl-SDbmLrwDysG1sTkvrfMhncjFSph5PzF3MITZW9aGMBfLrh19w</recordid><startdate>19980301</startdate><enddate>19980301</enddate><creator>Friedman, Sy D.</creator><general>Cambridge University Press</general><general>The Association for Symbolic Logic, Inc</general><general>Association for Symbolic Logic</general><scope>BSCLL</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>EOLOZ</scope><scope>FKUCP</scope><scope>IOIBA</scope><scope>K30</scope><scope>PAAUG</scope><scope>PAWHS</scope><scope>PAWZZ</scope><scope>PAXOH</scope><scope>PBHAV</scope><scope>PBQSW</scope><scope>PBYQZ</scope><scope>PCIWU</scope><scope>PCMID</scope><scope>PCZJX</scope><scope>PDGRG</scope><scope>PDWWI</scope><scope>PETMR</scope><scope>PFVGT</scope><scope>PGXDX</scope><scope>PIHIL</scope><scope>PISVA</scope><scope>PJCTQ</scope><scope>PJTMS</scope><scope>PLCHJ</scope><scope>PMHAD</scope><scope>PNQDJ</scope><scope>POUND</scope><scope>PPLAD</scope><scope>PQAPC</scope><scope>PQCAN</scope><scope>PQCMW</scope><scope>PQEME</scope><scope>PQHKH</scope><scope>PQMID</scope><scope>PQNCT</scope><scope>PQNET</scope><scope>PQSCT</scope><scope>PQSET</scope><scope>PSVJG</scope><scope>PVMQY</scope><scope>PZGFC</scope></search><sort><creationdate>19980301</creationdate><title>Generic saturation</title><author>Friedman, Sy D.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c337t-a85ab0d33aaf5ab31d39f473b6474af0ff66f064e69705909772637af7deff9e3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1998</creationdate><topic>Cofinality</topic><topic>Indiscernibles</topic><topic>Mathematical theorems</topic><topic>Periodicity</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Friedman, Sy D.</creatorcontrib><collection>Istex</collection><collection>CrossRef</collection><collection>Periodicals Index Online Segment 01</collection><collection>Periodicals Index Online Segment 04</collection><collection>Periodicals Index Online Segment 29</collection><collection>Periodicals Index Online</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - West</collection><collection>Primary Sources Access (Plan D) - International</collection><collection>Primary Sources Access & Build (Plan A) - MEA</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Midwest</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Northeast</collection><collection>Primary Sources Access (Plan D) - Southeast</collection><collection>Primary Sources Access (Plan D) - North Central</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Southeast</collection><collection>Primary Sources Access (Plan D) - South Central</collection><collection>Primary Sources Access & Build (Plan A) - UK / I</collection><collection>Primary Sources Access (Plan D) - Canada</collection><collection>Primary Sources Access (Plan D) - EMEALA</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - North Central</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - South Central</collection><collection>Primary Sources Access & Build (Plan A) - International</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - International</collection><collection>Primary Sources Access (Plan D) - West</collection><collection>Periodicals Index Online Segments 1-50</collection><collection>Primary Sources Access (Plan D) - APAC</collection><collection>Primary Sources Access (Plan D) - Midwest</collection><collection>Primary Sources Access (Plan D) - MEA</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - Canada</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - UK / I</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - EMEALA</collection><collection>Primary Sources Access & Build (Plan A) - APAC</collection><collection>Primary Sources Access & Build (Plan A) - Canada</collection><collection>Primary Sources Access & Build (Plan A) - West</collection><collection>Primary Sources Access & Build (Plan A) - EMEALA</collection><collection>Primary Sources Access (Plan D) - Northeast</collection><collection>Primary Sources Access & Build (Plan A) - Midwest</collection><collection>Primary Sources Access & Build (Plan A) - North Central</collection><collection>Primary Sources Access & Build (Plan A) - Northeast</collection><collection>Primary Sources Access & Build (Plan A) - South Central</collection><collection>Primary Sources Access & Build (Plan A) - Southeast</collection><collection>Primary Sources Access (Plan D) - UK / I</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - APAC</collection><collection>Primary Sources Access—Foundation Edition (Plan E) - MEA</collection><jtitle>The Journal of symbolic logic</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Friedman, Sy D.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Generic saturation</atitle><jtitle>The Journal of symbolic logic</jtitle><date>1998-03-01</date><risdate>1998</risdate><volume>63</volume><issue>1</issue><spage>158</spage><epage>162</epage><pages>158-162</pages><issn>0022-4812</issn><eissn>1943-5886</eissn><abstract>Assuming that ORD is ω + ω-Erdös we show that if a class forcing amenable to
L
(an L-forcing) has a generic then it has one definable in a set-generic extension of
L
[
O
#
]. In fact we may choose such a generic to be
periodic
in the sense that it preserve the indiscernibility of a final segment of a periodic subclass of the Silver indiscernibles, and therefore to be
almost codable
in the sense that it is definable from a real which is generic for an
L
-forcing (and which belongs to a set-generic extension of
L
[
0
#
]). This result is best possible in the sense that for any countable ordinal α there is an
L
-forcing which has generics but none periodic of period ≤ α. However, we do not know if an assumption beyond ZFC+“
O
#
exists” is actually necessary for these results.
Let
P
denote a class forcing definable over an amenable ground model 〈
L
,
A
〉 and assume that
O
#
exists.
D
efinition
.
P
is
relevant
if
P
has a generic definable in
L
[
0
#
].
P
is
almost relevant
if
P
has a generic definable in a set-generic extension of
L
[
0
#
].
R
emark
. The reverse Easton product of Cohen forcings 2
<κ
, κ regular is relevant. So are the Easton product and the full product, provided κ is restricted to the successor cardinals. See Chapter 3, Section Two of Friedman [3]. Of course any set-forcing (in
L
) is almost relevant.
Definition.
κ
is α-Erdös if whenever
C
is CUB in κ and
f
: [C]
<ω
→ κ is regressive (i.e.,
f
(
a
) < min(
a
)) then
f
has a homogeneous set of ordertype α.</abstract><cop>New York, USA</cop><pub>Cambridge University Press</pub><doi>10.2307/2586594</doi><tpages>5</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0022-4812 |
| ispartof | The Journal of symbolic logic, 1998-03, Vol.63 (1), p.158-162 |
| issn | 0022-4812 1943-5886 |
| language | eng |
| recordid | cdi_projecteuclid_primary_oai_CULeuclid_euclid_jsl_1183745464 |
| source | Jstor Complete Legacy; Periodicals Index Online; JSTOR Mathematics & Statistics |
| subjects | Cofinality Indiscernibles Mathematical theorems Periodicity |
| title | Generic saturation |
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