(Amazingly enough, DNS is one of the central points in the FCC's argument that ISPs provide an information service.)
Yes, in theory ISPs can outsource it, in theory a company can outsource everything. That doesn't make any difference to arguments about whether it's an integral part of the service though.
There are plenty more falsehoods in the order. The order states that DNS is analogous to a gateway server that translates addresses and not analogous to a directory service. The order claims that transparent caching is a critical aspect of ISP service that users have come to depend on, and dismisses comments pointing out that numerous modern web standards break transparent caching. The order states that the service an ISP provides is a "multi-user computer server" through which consumers access the Internet and ignores the fact that most consumers receive public IP addresses and are technically connected directly to the Internet (and that, technically, it is possible to host edge services using a broadband connection, even if doing so is rare). You can go read the order if you want more examples.
In effect I'm pretty certain that the vast majority of ISPs are running their own DNS servers. Making this point rather unimportant.
Any other technical inaccuracies?
They also suggested the DNS is like a proxy server (in their words, a "gateway") rather than a directory service. Does that count?
"Any other technical inaccuracies?"
That consumers continue to rely on transparent caching and that caching is a core ISP service. The FCC dismissed comments pointing out that TLS breaks transparent caching on the basis that there are websites that do not use TLS.
The order also claims that because people are able to access websites via an ISP network, the ISP provides people with the capability of whatever those websites do (e.g. under the order's reasoning, Verizon is providing me with the capability to have this conversation with you). You can argue that is an opinion and not a fact, but the order does not apply it consistently; for example, it does not assert that a phone company is providing an information service by virtue of its customers' ability to use a dialup ISP.
The order claims that by connecting to your ISP's network, you are receiving, "...computer access by multiple users to a computer server...that provides access to the Internet." Maybe that is just how the FCC interprets routers, but again it is not being consistently applied e.g. to the phone system.
If you want more, go read the order; the technical analysis is not very long.
I'm not aware of how much caching is used, but to my knowledge is not that common as it would introduce a lot of problems for developers. This is also wrong.
How much of their arguments are based on DNS and caching?
That is an outright falsehood.
"Joint Committee on Taxation (JCT), Congress's nonpartisan scorekeeper, predicted that the Senate tax bill would add about 0.1 percent more a year to growth over the next decade, far less than what Treasury says. JCT took into account the economic effects of the tax cuts on individual and business taxes, but not other policy changes advocated by the administration, such as welfare reform. The JCT says the Senate bill's total cost would be $1 trillion after considering growth effects.
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